The Elder Scrolls series come to mind. While not explicitly furry (in the sense that it features exclusively anthropomorphic animals), it does feature the Khajiit (anthropomorphic cats), Argonians (reptiles), and in Skyrim, intelligent dragons. Every race has its distinct, fleshed out culture, belief system, and appearance. My favorite is The Elder Scrolls Online, since each race gets fleshed out, rather than the beastfolk getting tacked-on as an afterthought.
In a similar vein, the Wizardry series 6 through 8 have plenty of furry races, including the Felpurr (cats, with excellent stats to the point of being OP), Rawulf (wolves and dogs, clerics), dragons, lizardmen, and even yeti/bigfoot.
Master of Orion is Master of Orion, with avians (Alkari), cats (Mrrshan), Soviet bears (Bulrathi), ants (Klackon), etc. My favorite part is playing as the Mrrshan and putting humanity to the sword - or better, conquering Earth and cleansing it of humans by resettling the billions across the galaxy on volcanic prison planets...
Ahem.
Age of Wonders features the Tigrans as one of the playable species (and really, I'd write more, but I just distracted myself with the awesome Tigrans of AoW III...)
