If you insist on using anthropomorphized animals, there are really only two ways to go...
1. EVERYONE is an anthropomorphized animal. There are no humans, elves or dwarves. No dragonborn or tieflings...well, I guess, maybe, there could be dragonborn. But just all tribes to civilizations of animal-people.
2. No more than 2. If you're using the PHB races, swap out two of the options for two animal-based races. The majority of "animal-people" walking around in the fantasy world are for NPCs, friendly -though generally not- or otherwise. Lizardmen, hyena-men, were-wolves-rats-bears-cats-etc-etc.. They're not meant to be PCs. So pick two options, give them an interesting story/place in the world, and add them to the player's choices.
Otherwise, and I think the largely preferred game/world/setting consistency, imhe and o, is to keep them exclusively monsters/NPCs and not allow them for PC races.