Hmm.... I cant think of what this Mortal Kombat one might be.
The only mode I'm aware of that isnt some sort of arena-based fighting like the normal fights is actually not a "mode", but a whole seperate game. That funky one, that was like a platformer/brawler, which starred Sub-Zero as the player character.
That being said, the recent MK games have ALOT of extra types of fights to do, where rules and conditions are very different from the norm, and stuff like that.
As for other games doing this.... it seems very rare among fighting games. I think most developers prefer to use their time and resources to work on the main modes, rather than create a strange side-mode that many players will often ignore. The only other game I can think of that did this at all was Guilty Gear Isuka, and that one..... ehhhhh. I consider it the black sheep of the series; the only game in that series that just wasnt much good at all. It had major control issues, among other screwball problems.
Well, there's also Ergheiz, an older (and very strange) fighting game featuring alot of characters from the Final Fantasy series and such. It had a bizarre, dungeon-exploring mode that was very, very different from the fighting modes (it played NOTHING like the fighting modes at all, from what I remember). Bloody weird game, that.