It's interesting. The big difference with the Nolanverse, besides it being significantly more grounded and contained, is Bruce sees his crusade as winnable, whereas most other versions he is pretty clear that it's a very abstract goal. It comes down to how the supervillains are justified. In the Nolanverse, there aren't really supervillains outside of the League and the Joker, nor are there other superheroes; of course in most other versions, Batman is just one of many costumed people.
So I don't know if the obsession is different, but the way it manifests is different. I think "obsessive" is one of the universal traits depicted in every version of Batman, even 60s TV Batman. In TDKRises, Bruce not being Batman is corrosive to his soul, similar to DKReturns. It's his obsessive destructive qualities turned inward instead of outward.
I think it's the nature of Bane's character to invite the story question of Batman's ending. Or the other way around — thinking of a "last Batman story" would seem to invite an omega character like Bane.
Anyway it's just interesting the way that different writers have approached the question of "Can Batman be happy and be Batman?" with different answers, but I think only Tom King's version answers "yes."