
Dogs
Kris is compared to a dog multiple times. I think everyone has noticed this by now because it happens so many times. They even have a dog suit.
Susie tells Kris they'll have to wear "doggy ears" and do her homework for her since Ralsei can't, which implies Ralsei's ears are "doggy ears" (at least in Susie's opinion.)
Both Kris and Ralsei can have the annoying dog appear in their stats. For Kris, it replaces the first blank ??? slot and for Ralsei, it replaces the Kindness slot (with "Dogness".)
Kris, Ralsei, and Susie all have dog sprites when ACTing against Tasque Manager (even though Tasque Manager is a cat...?) Susie is notably poorly behaved as a dog; Tasque Manager commends the "Beware of Susie" signpost for its obedience rather than Susie herself. This ACT is called "Order," and the flavor text says Tasque Manager "shows you order," so dogs are definitely linked to order. Tasque Manager says Ralsei "really loves to do what he's told."
Kris and Ralsei are the two party members most concerned with carrying out the prophecy and they also both railroad the player.
The Wet-Nose Bandits, the dog-burglar group, notably interrupt your attempt to warn Undyne about the dark worlds with their jailbreak, thus unintentionally ensuring the plot continues as normal.
The Annoying Dog railroads you at many points in chapter 2: it causes the traffic jam that forces you into the library, it carries you through a puzzle in Queen's mansion, and, most blatantly, a literal roadblock of traffic cones with the Annoying Dog's face on them randomly spawns to force you into the Noelle-only section.
The Annoying Dog is obviously a stand-in for Toby Fox, the creator of Undertale and Deltarune. (Shocking revelation!) A dog is the God of this universe.
The Annoying Dog appears in chapter 2 to run you over when you first try to enter the egg/moss room. Some unused assets in chapter 4 indicate the Annoying Dog climbing section may have been linked to the chapter 4 egg at some point. Maybe the Dog has an "issue" with the eggs?
"HEAR THAT WHINE!? THAT'S YOUR CRT ASKING FOR A WALK!!" Tenna is dog-coded. Tenna's role in chapter 3 is just to put down roadblocks and stall for time so things can progress as intended. Tenna also parallels with Kris, a dog-coded character. Tenna does appear as a cat-like sphinx, but sphinx-Tenna gives you one of his fake noses that he stole from Spamton, so the cat theming is also likely stolen from Spamton.
ICE-E's Japanese name is Bulldokkun, a pun on "bulldog," so he is most likely a dog.
https://toby.fangamer.com/interviews/nine/
Seven is a cool dog-or-wolf that five sics on eight.
Cats
The Puppet/Cat elemental pair exists. Tasques, Tasque Manager, and the cat Shadowguys in Rouxls' chapter 3 battle all have Puppet/Cat attacks. So does Spamton NEO.
The Mannequin, Frayed Bowtie, and Dealmaker armors protect against Puppet/Cat attacks. Only Kris can equip the Mannequin. Noelle says the Frayed Bowtie reminds her of Asgore.
FRIEND is a cat. FRIEND's eyes are the yellow and pink of the default and optional alternate SPARE colors in Undertale. FRIEND appears in the basement of Queen's mansion, in hidden rooms in the first round of the Sword Route, in the boss fight of the Sword Route, and in a hidden room in the 3rd Sanctuary. The hidden room has a chest with a Waferguard, the armor sold in Gerson's shop. Since the Sword Route gives an armor that helps with Chapter 3's Shadow Crystal boss, that means most in-game FRIEND appearances are linked to a Shadow Crystal boss.
FRIEND also appears as a rarecat in the Battat minigame, where it can either decrease or greatly increase your points. Truly, FRIEND can be your devil or your angle. Notably, the FRIEND in this minigame has a different shade of pink for one of its eyes. Super angel cats also appear in this minigame.
FRIEND is commonly thought to be linked to the SMILE option in the S-rank room vending machine. That option is linked to the "Always at your humble service" message, as both disappear at the same time, which ties back to Undertale's hidden demon messages where the demon refers to themself as your "humble servant."
The Odd Controller essential to the Sword Route is noted to have pink and yellow buttons. It is most likely related to the "knock-off" controller under Asriel's bed. This may be a reference to the Mad Catz brand of knock-off controllers.
Seam is a cat. Seam is highly linked to the Shadow Crystals, far more than FRIEND is. While cynical about whether it will amount to anything in the end, they highly encourage the player to do the optional Shadow Crystal content. Seam has a lot of weird shit going on in general.
Seam's name in Japanese is Nui (TL note: this means "stitch".) When they introduce themself, instead of saying "it's pronounced Shawm," they say you can read it backwards as "Inu" (TL note: this means "dog".) Then they say that since they own a shop, they're better off than a dog in terms of social standing.
Seam's Light World counterpart is in a classroom that used to belong to Gerson. In the 10th anniversary, Gerson speaks fondly of one of the human magicians who sealed away the monsters. Seam is a magician who sealed away their companion, Jevil. Seam and Gaster both have only one eye open. "What do you call a man with one eye?" "In this land, only eyes blinded by darkness can see the way..." https://en.wiktionary.org/wiki/in_the_land_of_the_blind,_the_one-eyed_man_is_king
There may be a Seam-Gerson connection analogous to the FRIEND-Gaster connection. Gerson and Gaster both had their roles filled by Alphys in DR and UT respectively, Gerson took Gaster's spot in the official Undertale tarot deck, Gerson is the author of the Lord of the Hammer rumored to have fell into his own creation just Gaster did, Gaster is also likely the creator of the prophecy Gerson based his story on, and they are both green.
Seam is noted to have "lost their smile" since sealing Jevil away, a link back to the ever-smiling FRIEND. Seam appears in Asriel's desktop wallpaper, while FRIEND has a connection to the knockoff controller Asriel used. Seam had the Shadow Mantle in their shop at some point before it was taken to the TV world, while FRIEND is present in the Mantle holder fight. Both Seam and FRIEND have yellow-colored left eyes. Seam quotes Gaster's "dark yet darker" line from entry 17, while FRIEND appears in a cutscene where Ralsei references the entry 17.
Catti is a cat. She is linked to black magic (Satanic) and consorts with snakes (also Satanic.) She likes Noelle, a mouse-coded character. She has yellow eyes and pink hair. It's a brighter shade of pink similar to the one FRIEND has in the Battat minigame.
She is linked to the game Cat Petterz in the Sweepstakes, much like Spamton and FRIEND. She is specifically linked to the Cat Petterz RPG in the game itself. Noelle considers the Cat Petterz RPG a ripoff of Dragon Blazers, but it's unclear if she strictly means in terms of gameplay or if the Cat Petterz RPG is yet another retelling of the prophecy.
There is a horror movie called "Invasion of the Cat Petters" of unknown relation to Cat Petterz.
In Undertale, Mad Mew Mew and Mettaton both have cat theming. Mad Mew Mew is obvious, and notably she is also pink and yellow. Mettaton is pink as well, but closer to FRIEND's Battat minigame pink than regular FRIEND and Maddy's pastel pink. Mettaton's dress reappears in Deltarune attached to the Mannequin that serves as Puppet/Cat armor. Both Maddy and Mettaton's vessels are puppet-like.
I've always thought the horns on the Titan here look like cat ears:

Overall, cats seem to be more aligned with chaos and optional content, compared to dogs' association with order and railroading. Cats are associated with both demons and angels, similar to how the SOUL is associated with both the angel doll and the "demon heart" in Raise Up Your Bat. Cats are also associated with puppets.
Mice
Noelle is a mouse. That's why she says stupid shit like "Excheese me?" Noelle is a parallel for the player: she enjoys seeking out secrets in games, might be the Angel, and she's a huge gay nerd just like YOU. Haha, got your ass.
Noelle has connections to multiple cat-themed characters, such as Spamton, Catti, and FRIEND.
Noelle overcomes her shyness as she becomes less afraid of mice, symbolized by her Boldness stat.
The mouse puzzles are pivotal for the weird route.
Noelle's reaction to the mouse babysitter NPC also corresponds with her attitude towards the player in the normal route versus the weird route.
Jevil is a mouse. He lives in a mousehole and had cat-and-mouse games with Seam. Jevil is also a parallel to the player because he sees the world as a game.
Noelle and Susie mistake the SOUL under the angel doll for either a mouse or a severed hand. The Maus turns into a severed hand so you can use it in the Mike minigames.
The puzzle piece guys who give battle tutorials in chapter 1 are all Jigsawrys, meaning they are MOUSE:PUZZ element. They have substantial meta knowledge, due to their tutorial role. The donation hole may count as a mouse hole.
Jigsawry's recruit info is "It cries tears of joy that you are its boss." Mauswheel's recruit info is "You RECRUITed 3 big mice and you're stuck with it." Tasque's recruit info is "This cat loves you!" Most recruit info does not directly reference you, the player, so these three are all notable in this regard. Also, Maus's recruit info references Tom and Jerry, a show that's mainly about a cat and a mouse with a mutually antagonistic relationship (toxic yaoi), but also includes a dog as a third party in their conflicts. Usually the dog sides with the mouse, but sometimes the cat and the mouse team up against the dog.
The recurring mouse hole savepoints in Undertale symbolize the player: the mouse is ever-present, but never seen. It only has a tangible effect on the world in the pacifist epilogue. The mouse holes have different items nearby throughout the game: first the cheese is simply on a table, then the spaghetti is on a table and frozen with an unplugged microwave next to it, then the cheese is encased in a magical crystal, and finally the cheese is locked inside a computerized safe. The mouse somehow obtains the cheese/spaghetti in all cases in the pacifist epilogue.
You can tell Onionsan that your name is either "Kris" or "Hippopotamus," which is a reference to Mega Man Legends. Onion misremembers "Hippopotamus" as "Hyper-potato-mouse" which he subsequently shortens to just "Mouse." Thus it's a choice between being identified as a Kris or as a mouse.
I'm cheesed that you read this far.
Snakes/Lizards
This will be a bullet point section since it's more of an extension of the cat motif.
- Susie
- Jockington (snake) is friends with Catti (cat)
- Jockington's room in Queen's mansion used to belong to Spamton (cat)
- Lizard enemy in the Sword Route game
- Bratty (lizard) is frenemies with Catty (cat)
- Alphys (lizard) has a pet cat who is actually Susie (lizard) and also Catty (cat)
- Alphys (lizard) is friends with Mettaton (cat-adjacent)
- DRAGON Blazers vs CAT Petterz
- "The darkness gives a long gaze, which slithered like a snake."
- "O, the shadow will grow, serpentine!' [...] "A patch of shadow, slithering on, slithering on, serpentine!"
- "For me, she Covered in Hair Always. We Do What, the other can n't!" (Jockington grows a beard = Jockington is covered in hair)
- Something about Susie's fixation on apple shampoo and the whole biblical snake-apple thing. I don't know.
- "a cat's tail is just a snake in disguise"

- Undertale concept art for the Snowdin Forest area includes a mural with a dragon fighting an entity with angel wings, snakes, and the Deltarune
Other
Ralsei's monster movie costume is either a green bunny or an eyeless frog. Rabbick says, "Bunnies are the sequel to frogs," so it may be both. Rabbick's attacks mimic Froggit's, and in turn, Ribbick's attacks mimic Rabbick's. Rabbits are housepets outside the dog/cat dichtomy, but without mice's vermin association. Frogs are amphibians, the un-lizards. There are connections between snakes, frogs, and slugs/snails in Japanese mythology. Mew Mew and boss monsters both eat snails...?
Boss monsters, aside from their goat-like qualities, have some ambiguous traits that could make them either dog, cat, or bunny-coded, depending on your perspective.
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https://rentry.org/elementalpairs