Some more assorted games:

Leo the Lion and Leo the Lion's Puzzles seem to fit, having a lion as the titular character.

Tanglewood is a platformer with a fox-like creature as the main character.

Pushover is a puzzle with an anthropomorphic ant as the protagonist.

The Next Big Thing features an appearance by a human-fly hybrid person, Professor Fly.

Thank you all for the suggestions! I'll add these games to the list:

Quest for Glory 1-5
Bug Fables: The Everlasting Sapling
King's Quest 5
King's Quest 7
Toonstruck
Wallace & Gromit's Grand Adventures
The Alliance Alive HD Remastered
Octodad: Dadliest Catch
The Elder Scrolls III: Morrowind
The Elder Scrolls IV: Oblivion
Elder Scrolls: Arena
Elder Scrolls Chapter II: Daggerfall
Ultima 4+5+6
Ultima 7 The Complete Edition
Ultima Underworld: The Stygian Abyss
Bear With Me: The Complete Collection
RiME
Mutant Year Zero: Road to Eden
Detective Gallo
Chicken Police
Ancestors: The Humankind Odyssey
SpongeBob SquarePants: Battle for Bikini Bottom - Rehydrated
Chook & Sosig: Walk the Plank
Night in the Woods: Weird Autumn Edition
Alba: A Wildlife Adventure
Earthworm Jim 1+2
Earthworm Jim 3D
The Manhole: Masterpiece Edition
Dust: An Elysian Tail
Kero Blaster
Legend of Kay: Anniversary
Leo the Lion
Leo the Lion's Puzzles
Pushover
The Next BIG Thing

I have doubts for some games because they seem too far from the initial definition, and somewhere I have to draw a line or I should add any game that contains monsters even vaguely similar to Earth animals and this would include probably hundreds of other titles.

It's very interesting to note that some fantasy worlds are almost identical to Earth, but with talking animals, while others have creatures clearly inspired by Earth but have a precise biological identity in their own right. In some cases I dare say that the line between fantasy and sci-fi is really blurred. It's likely that a thorough knowledge of the lore of a game (world in which it takes place, origin of the species being analyzed) may lead to the inclusion or exclusion of a game from the list.

But I would like to know what you think about the excluded titles and also possibly those already included, in case you have more in-depth knowledge of the games in question.

Titles excluded for now:

Fell Seal: Arbiter's Mark - It seems like a game populated by so many creatures, but none seem to have any real connection to Earth animals.
Giants: Citizen Kabuto - There seems to be more monstrous creatures or aliens here that are not closely related to the animals we know on Earth.
Niche - a genetics survival game - The creatures in the game are supposed to be full-fledged aliens if I'm not mistaken.
Rayman's Raving Rabbids - This is a tough one, Rabbids resemble rabbits but are in fact different creatures, so I would rule that out.
Owlboy - I think the owl men in the game don't have a real connection to the owls we know, and are more associated with an alien world and culture.
Ori and the Blind Forest - Does this game feature real Earth animals?
Tanglewood - It seems that the main character is never really referred to as a fox and the resemblance seems more of a coincidence, in addition to the fact that everything seems to take place on an alien planet.

More or less, yes. They resemble animals so could be a possible candidate is why I suggested it.

Some titles included I wouldn't count as being about animals, The Alliance. Human looking characters with a tail and animal ears don't count as animals either. Including this Ne no kami could be in as well. Or practically any RPG in which your characters can transform into animals.

Here is another one for the list Megaquarium

About The Alliance, according to dtgreene's description one of the party members is a real penguin, can you confirm?
Also, are the beastfolk enemies more animal-like or also just with animal ears and tails?

Megaquarium is a great addition to the list. And I would also add Fight'N Rage, in which one of the main characters is a minotaur and all the enemies are mutant animals.

I can confirm this. Take a look and decide for yourself Robbins ;-)

I must still hold that if this one is in, you could also consider Gibbous A Cthulu Adventure since you are able to control Gibbous during certain parts of the game. :-)

Albion

The Iskai resemble cats, see:
https://www.gog.com/game/albion

or here:
https://albionworld.fandom.com/wiki/Iskai

At some point you have Iskais in your group.



Wing Commander series

The Kilrathi also resemble cats, see:

https://wingcommander.fandom.com/wiki/Kilrathi




Wizards & Warriors (2000)
https://www.gog.com/game/wizards_warriors

You can see in the screenshots that you can play several races: Ratling, Lizzord, Whiskah (cats), Gourk (pigs), Omphaaz (elephants)

Yes, I would say The Alliance can stay on the list and Gibbous is a fitting addition, thank you!

Iskais and Kilrathi are aliens, so I would exclude them according to the rules I set earlier, but they will be perfect for the next topic on alien creatures.

Wizards & Warriors instead it seems that it has exactly the right requirements, thanks!

I've also edited the rules slightly to make it clearer what I mean; it's harder than it sounds to provide rules that are stringent enough to not include every game with vaguely terrestrial races and creatures but include games with anthropomorphic animals, which make up the majority of games.

I remembered some other games:

Similar to the Wizardry series and Wizards & Warriors:
Grimoire : Heralds of the Winged Exemplar
https://www.gog.com/game/grimoire_heralds_of_the_winged_exemplar

http://grimoire.wiki/index.php/Races
--> Wolfin, Saurian, Naga, Rhattu, Leonar

Crow is a talking bird and the sidekick for several main characters in The Longest Journey/Dreamfall/Dreamfall Chapters
https://tlj.fandom.com/wiki/Crow
https://www.gog.com/game/the_longest_journey
https://www.gog.com/game/dreamfall_the_longest_journey
https://www.gog.com/game/dreamfall_chapters_season_pass

You play a wolf in Lost Ember:
https://www.gog.com/game/lost_ember

You play a bird in A Short Hike
https://www.gog.com/game/a_short_hike

Beavers in Timberborn:
https://www.gog.com/game/timberborn

Beautiful list. Rusty Lake games are generally well populated with animals, but Rusty Lake Hotel in particular has anthropomorphic ones as characters.
In mostly all RL games there is some animal to interact with. In Blasphemous you can pat a dog on the head in the town. :)
That's not much.

Rain World has futuristic imaginary animals adapted to the environment of the future. Up to you to say if a slugcat is an animal or a cute monster.
But the creatures in the game beheacve like a smart system interacting with each other like animals in nature, hunting and such.
In Reigns there are characters like two birds and a dog.
The Count Lucanor has quite the amount of goats, and other animals.

The Aquatic Adventure of the Last Human has plenty of oversized sealife, like giant seahorses and such.
Abzu is actually quite dedicated to prehistoric sealife and you can do fishwatching.
In Downwell you can crush frogs, bats, turtles, snakes and worms under your boots or your boot-bullets.
In Stasis there is a little scare jump with a black cat, that's also mentioned in diaries that can be read :)
In Anna's Quest there are many animals, like the whole Town Musicians of Bremen band, looking for a new place to stay. Real and stuffed talking bears, goat, raven, actually many.

Little Big Adventure had many anthopomorphic animals, like elephants, and bunnies, called grobos and rabbibunnies.
In Slain you can ride a big wolf spirit, I don't remember what other animals are in game.
Sorry for the multiple post! I know many of the ones I have mentioned don't fit the list probably but I was having fun thinking about those games. I think the best ones would be:
Little Big Adventure
Anna's Quest

I would argue that Boo wouldn't count, because, from a game mechanics perspective, he's just an item that does nothing (except display a message if you attempt to unequip him), rather than an actual character.

With that said, Baldur's Gate 2: Enhanced Edition apparently features a recruitable bear (Wilson).

I see there are a lot of great new candidates, thanks again to everyone for the suggestions and for the accurate descriptions.
All of these games seem to fit, and Crow seems like a very important character for the games he appears in, so I'm adding them all to the list.