I don't know if you're relegating each race to a specific class, rather than a role, out of roleplaying, or aestethic reasons, but I think the important pairing is race/role. Of course, you're free to play whatever combination of those you want, I myself have a bunch of non-meta combos.

In my opinion:

• Argonian: any healer or tank class.
• Dunmer: pretty much anything, maybe just not tank.
• Nord: tank, kinda.
• Altmer: mag dps or healer.
• Bosmer: gank PvP (maybe, I don't play PvP, but that seems the most appropriate role for them, they do seem like a PvP-focused class anyways).
• Khajiit: same as Dunmer.
• Breton: healer, or tank in some situation.
• Orc: stam, or even mag dps now, tank.
• Redguard: they are pretty bad right now, but they are ok stam dps and tanks.
• Imperial: tank.

In general, I would say pretty much any race that has even some sort of passive that aids with magicka or spell damage/crit is viable for healer, even though Argonian and Breton are probably the better ones.
Now, do take this with a massive grain of salt, but I think with tanks race is not as important as with other roles. It can make a difference in harder content, but I have seen people tank with Bosmer or Khajiit and do well in pretty much anything (just like I've seen top-tier Argonian DDs). This is a personal belief of mine, even though it's a capital sin of endgame play, I think skill trumps any passive, even though I would still recommend the usual race/role suggestions as they will make the role easier to play overall.