The Elder Scrolls games are all about choice, you pick WHO you wanna be and play how you wanna play, however it is my belief that certain games had certain races that made more sense to play, and I'm curious if people would agree
Morrowind: Dunmer or Altmer
Dunmer: The Story of Morrowind is set in non-other then..Morrowind, mainly VVardenfell the Dunmer Homeland. And since you are the reincarnation of Nerevar who was a chimer, and the Chimer race became the Dunmer, it only makes sense.
Atlmer: Same reason as above mainly, the Chimer were at one point Altmer, but they split apart and left the Summerset Isles thus becoming the Chimer.
Oblivion: Imperial or Nord
Imperial: Oblivion takes place in the Seat of the Empire, the very capital of Tamriel. Thus the Champion of Cyrodill being Imperial makes sense.
Nord: Imperial and Nord history are so inter-mingled a Nord Champion would also be plausible, though more so Imperial
Skyrim: Nord or Dunmer
Nord: The game takes place in Skyrim, and the Nord people were the ones who lived under Dragon Rule, so a Dragonborn being Nord only makes sense
Dunmer: Kind of a stretch, but with the RED YEAR many Dunmer relocated to Skyrim thus a Dunmer Dragonborn coming to Skyrim would be plausible, though a Nord would make more sense.
Daggerfall: Breton
Breton: Daggerfall takes place in High Rock, the Breton homeland, logically when you send an agent to a place you send someone who will blend in. So (to me) the Emperor sending an Agent of the Breton Race would make sense
***Please Note I only loosely know the story of Daggerfall so I may be wrong about this***
Please know this is just my personal opinion, others are welcome Arena and Battlespire I know next to nothing about. And Redguard you dont pick your race.