Often when I'm feeling dogmode or any other kind of alterhuman way, I like to play video games as a way of self-expressing. Especially ones that let me play as something similar to [x creature] I feel like at that moment. Here are some I'd recommend to others, sorted into vague categories. Almost all of them will include some degree of character customization, as that's what I look for the most in games like this.
WolfQuest: A simulation game where you live as a wolf, finding a mate and raising pups together. There's a learning curve, but it's so much fun to play. The devs really do a wonderful job of striking a balance between realism and enjoyable gameplay.
WolfHome (browser game): An MMO/chatroom that is very old and very dead, but I still like to hop in it and emote for funsies.
Reach of The River: You can play as a deer or a wolf.
Bunhouse: A cute gardening game where you play as a rabbit that can walk on both two legs and four.
The Deer God: A pixel game on Steam where you get to be a deer. Not super great, but I liked how relaxing it was.
The Endless Forest: A free indie game from the early 2000s where you play as a deer with a human face in an idyllic forest. No chat system, no identification beyond a pictogram if you choose to make an account. You can customize your deer with masks, pelts, and antlers, but you need the help of other players to do it!
Reach of the River: You can play as a deer or a wolf.
Feather: A relaxing game where you play as a bird and fly around. Yes, that's it. Yes, it's beautiful.
Untitled Goose Game: It's a beautiful day, and you are a horrible goose.
Various real-world animals:
Meadow: A relaxing, no-combat, no-chat game where you can play as various mammals (and a few birds, and a frog!) including lynxes, foxes, goats/deer, rabbits, hedgehogs, pheasants...
The Shelter Series (including Paws): Badger (Shelter), Lynx (Shelter 2/Paws), Elephants (Shelter 3). Made by the same company that made Meadow.
Animalia Survival: An MMO survival game where you can play as a lion, zebra, wildebeest, hunting dog, and various other African animals. This one isn't as enjoyable as a game, it relies heavily on finding a good server that isn't toxic, but I still like prowling around as a lioness in it.
Beasts of Bermuda: Like Animalia Survival, but with dinosaurs. Relies on finding a good server.
Lil Gator Game: You are a baby crocodile anthro running around in a world of other creatures.
Empires of the Undergrowth: Ant life simulator. It's pretty good, but strategy-oriented.
Rain World: Slugcats are both slugs and cats. Somehow they feel more like aliens to me though.
Outer Wilds: You are an alien exploring space. This one has a beautiful story and is very immersive.
Drunk on Nectar: Various bugs, including jumping spider, mantis, and butterfly.
Warframe: A free game that has a fantastic story once you get into it. Lots of shooting things in space, but you can swap between Warframes and customize your Warframe's colours etc.
The Talos Principle: A puzzle game with strong themes of religion, autonomy, personhood, etc. You are a robot.
Book of Travels: If you kin any kind of human from older eras of humanity you might like this one. I can't explain it, but it brings me comfort when I'm feeling Paleolithic despite not being in that era at all.
Roblox: There are tons of games on Roblox that will allow you to play as all kinds of creatures, from realistic to fantasy. Warrior Cats Ultimate Edition, Feather Family, and Farm Life are ones I enjoy a lot. There are various dragon and dinosaur ones too, gryphons, unicorns, etc.
Furcadia: There are so many realms where you can be whatever you want. The community is very furry and alterhuman friendly. It can be quiet in the SFW areas but it's still absolutely worth checking out.
Illarion: You can play as various fantasy races. Roleplay is enforced, but I personally like it for wandering around as an orc or a lizardman or what have you.
Ancestors: The Humankind Odyssey: If you kin any kind of ape or prehistoric human, this one's for you!
Personally I really recommend checking out Roblox, there are SO many beautifully-made games with shockingly good graphics on there. Not just for therians either!
For humanoids, I also generally recommend games with fun character customization screens like Skyrim, Baldur's Gate 3, and Elden Ring, even if the gameplay itself doesn't bring me much euphoria.
Please feel free to add your own recommendations and thoughts!