So what happens when multiple deadra lords lay claim to a single mortal's soul? Doing all of their quests for two straight games have made me curious.

none of the daedra (excluding nocturnal and possibly sithis) have ever claimed the players soul

Sithis is a deadra? Also doesn't Malog Bal take people's souls?

Sithis ain't a Daedra. Sithis is nothingness.

By being a werewolf, the player is claimed by Hircine too.

The general consensus seems to be whoever has the strongest claim gets you. But we really don't know how it all works, either... it used to be that Arkay would prevent a mortal's soul from being used in a way against the person's will, but the idea of a Daedric Prince (or the Ideal Masters) being able to take a mortal soul even if the person never agreed kinda goes against that.

Add to that the player's unique circumstance of being Dragonborn, having the soul of a dragon granted by Akatosh, puts another spin on it all, too.

The Akatosh thing is a good point. So who's claim is stronger though, Hircine with Lycanthropy or Nocturnal as a Nightengale assuming the player does both.

Of course, the only reason the Ideal Masters can take the souls in the first place is because of Necromancer's moon, the main purpose of which appears to be subverting Arkay.
That said, the previous two games have both had the the PC becoming (nearly) immortal in some divine/daedric fashion. We've yet to see how Dovahkiin ends up. Unless you consider either the vampiric or werewolf forms of immortality as their end, which I don't.

Molag Bal

Hircine

Nocturnal

Mehrunes Dagon

Ideal Masters (Only if bound to a soul gem)