This is a short list, I'm just going to list the ones I'm familiar with. I'd love to hear about some more.

Ever Oasis (3DS)

The one that prompted this post. I just picked it up last week and I'm really enthusiastic about it. Even ignoring the anthro designs, this game is perfect for me - the best I can describe it is if you put Legend of Zelda and Animal Crossing in a blender, and made the result more adorable and polished.

Solatorobo (DS), Tail Concerto (PS1)

I only played Solatorobo. The common opinion seems to be that it's not a great game, but it's clear that a lot of love went into the world. If you care much for world building and interesting designs you might want to try these. They're part of LTB, an ongoing passion project by CyberConnect2 that isn't really profitable (not that it needs to be, they have the Naruto games for that). There's lots of supplemental material including three huge art books.

LTB differs from the other items on this list in that it's overtly furry. CyberConnect2 publishes unrelated anthro art books in Japan called The Kemono Magazine.

Pokemon

I used to have the perspective that fans took innocent designs and perverted them into R34. With the release of Salazzle in Gen 7, there's no longer any denying it - Game Freak know exactly what they're doing.

Obvious mentions

• Animal Crossing
• Undertale

Honorable mentions

• Legend of Zelda for all the fish and bird people
• Elder Scrolls for all the cat and reptile people
• Cave Story for all the bunny people

Fur Fighters
Dreamcast
PC
PS2
IOS

I mean I hate to be the one who has to say it, but...

Sonic?

--> Ōkami - There are anthro birds, speaking animals, anthro gods etc.
--> Star Fox - Should be known by most people by now (thanks to Krystal)
--> Tooth and Tail RTS game with anthropomorphic units
--> Sly Cooper - Antho characters all around, Camelita Fox has a lot of pics here
--> Conker's Bad Fur Day - Black humor with anthro characters
--> Spyro
--> Gex series)
--> Donkey Kong series
--> Super Monkey Ball series
--> Crash Bandicoot series
--> Earthworm Jim)
--> Bubsy series
--> S.W.I.N.E..
--> Hogs of War
--> Worms series)
That's all that comes to my mind currently

The bloody Roar series lets you turn into animals.

D4rk said:
--> Ōkami - There are anthro birds, speaking animals, anthro gods etc.
--> Star Fox - Should be known by most people by now (thanks to Krystal)
--> Tooth and Tail RTS game with anthropomorphic units
--> Sly Cooper - Antho characters all around, Camelita Fox has a lot of pics here
--> Conker's Bad Fur Day - Black humor with anthro characters
--> Spyro
--> Gex series)
--> Donkey Kong series
--> Super Monkey Ball series
--> Crash Bandicoot series
--> Earthworm Jim)
--> Bubsy series
--> S.W.I.N.E..
--> Hogs of War
--> Worms series)
That's all that comes to my mind currently

Ah... Furry video games of the 90's. :D

I"ll add in James Pond-- an anthro fish who platformed around trying to defeat Dr. Maybe. He had 2 sequels in the 90's... James Pond 2: Codename Robocod and James Pond 3: Operation Starfish.

and the Holy Grail as a kid for me: Inherit the Earth -- a point and click RPG adventure game, with a cast of anthro characters.

This was significant because a lot of other games had characters who were cartoony animal-characters... but very few had bipedal anthro animal characters

There was also "Brutal: Paws of Fury" -- was mortal ocmbat, but with furry characters. BUt with creative names like Kendo Coyote, and Tai Cheetah

Adventures of Dragon
The Pirate's Fate
Zeran's Folly
Changed
The Tenth Line
Pizza Express
Armello
Night in the Woods
Dust: An Elysian Tail
The Night of the Rabbit
Jolly Rover
Ty the Tasmanian Tiger
Don't Make Love
Bear With Me

...there's plenty more, but those are are the first that popped to my mind. Plus a couple of others that I don't want to mention, because of dev drama. First three contain sexual themes, and were blatantly made by furries. (First four? Haven't played Changed yet, but judging by the screenshots..)

Unrest has nagas, but you only get to play one for a short section. Not sure if that counts.

World of Warcraft, Guild Wars 2, Divinity: Original Sin 2, Morrowind, Oblivion, Skyrim, Avernum series, Starbound... will add more when I can think of them.

Here's a few:

• The Banjo Kazooie Series) is a pretty obvious choice. Most people know what it is so I won't give too much of an explanation; it's just a fun toony anthro focused platformer.
• Nod to the spiritual successor Yooka Laylee here too for obvious reason.
• Blazing Dragons) is a really charming point and click anthro game starring dragons filled with Monty Python-esque humor and charm. The game in itself was based on a cartoon of the same name (designs ingame are heavily reworked in comparison to the originals) which is considered a cult classic british and anthro cartoon in some circles and has quite the legacy.
• OneShot is a really fantastic indie puzzle/adventure game with charms and cute designs and characters rivaling Undertale's. You explore a world with a dead sun as a young, ambiguously gendered cat child named Niko as you try to restore the sun and save the world.
• Whiplash) is, in my personally opinion, a really charming and underrated classic platformer starring the duo (connected by the wrist via handcuffs) Spanx the weasel and a Redmond the rabbit escaping an animal testing company. The goal is to destroy as much as possible and cause as much monetary damage as you can, and to also free your fellow animals while destroying this capitalist hell that is the whole company. It's really fucking funny and has great humor and voice acting, and it is criminally underrated. The only issues with it in my opinion are some late game backtracking and the game is a bit buggy. Otherwise I really enjoy it.
• Gonna give a small nod here to Mad Dash Racing, an xbox launch title racing game featuring a huge cast filled with mostly anthro characters. Spanx debuted in this game before Whiplash came out, similar to Banjo appearing in DKR and then appearing in Banjo Kazooie later on.

Bloody Roar Primal Fury

is a pretty good one in hindsight.

Never really considered it until recently; Realizing the game is literally
a fighter where peeps turn into anthro dood to beat/claw the stuff outta each
other.╹‿╹)

Pretty good 3-D, tekken-like fighter imo
◠‿╹)

Ghost of a Tale
A grimdark stealth adventure as a mouse against a kingdom of rats and other creatures. Quietly solve environmental puzzles while dodging guards and monsters, or probably die of some horrible fate.

Moss
Third-person action/adventure for VR, also as a mouse. I own this, and I should really play this. It was maximum cute when they showed the main character using sign-language to introduce themselves in a marketing video, and I'm super curious to know whether they carried that idea into the full game.

Freedom Planet
2D side-scrolling platformer which started its life as a Sonic the Hedgehog fan-game, then went legit as their own IP. Might be a little too girly at parts if you're a dudebro.

I would plug Dust: An Elysian Tale again, a side-scrolling action game as a squirrel(?) swordsman, but I'm biased because I'm amazed it was almost entirely the work of one guy. You might recognize Fidget from posts here.

Honorable mention:
Ori and the Blind Forest
Though not specifically furry, it features a couple of anthropomorphs and will probably appeal to furry sensibilities. Also just simply gorgeous artwork in general.

Oh, and there's an official Redwall game, but I know almost nothing about it except that the developers come from a mobile gaming background, have no experience in the genre they're entering, and were not crazy-hardcore fans of Brian Jacques' works. So I'm cautious about this one, will personally wait to hear more before getting my hopes up.

Hollow Knight has anthro bugs.

Captain Claw anyone?

Fox n Forests
A sidescrolling platformer where you play as an anthro fox and magically change the seasons as part of dealing with enemies and obstacles.

Tunic
An upcoming Zelda-esque game where you play as an anthro fox.

Lucky's Tale
Also called "Super Lucky's Tale" on non-Oculus platforms. A 3D platformer for VR, where you play as an anthro fox, collecting coins and fighting villainy.