Blaster Master was known for this. And then you had the Wiiware Remake froma few years back.
The light Map-running segments in NES Bionic Commando really stuck out for me at the time. It was like just a small bit of strategy mixed with a sidescrolling shooter.... that suddenly had platforming. And then they threw in Commando, because... SURE, WHY NOT!!
Noone else will care, but I still think of Sega CD's Revengers of Vengence whenever I think of genre mashups. The Overhead RPG with a fighting game as it's main combat, effects by stats.... and then overhead shooting segments, because, once again, SURE, WHY NOT! I still want to see someone make a more informed, more workable balance of this concept (well, they can cut the shooter part.)
I also think Legend of the Mystical Ninja + The rest of the Ganbare Goemon game series count in unique ways. First we have the 3/4ths view town segments & then the sidescrolling action stages, and a LOT of mini-games such as painting, Gradius, 3D mazes, quiz games, etc. And then the later games started to add the Goemon Impact Mecha battles, or specific Mini-games per boss battle.
Sonic Adventure comes to mind outside of Big The Cat's fishing, but yeah I love how they gave the characters separate gameplay instead of them all being the same thing. Made it unique and fresh for me.
Also a game most people probably don't know about,
Mazin Saga Mutant Fighter for
SEGA Genesis.
The main game is a beat-em-up with very fluid animations that is brutally hard, but when you fight bosses it turns into a fighting game. Not a Capcom style fighting game, but one similar to but not quite to Smash Bros. It's pretty awesome. It's kind of like footsies in fighting games, though or fencing or even Punch Out with the way you fight. You have to learn the patterns and your moves and dodge or block accordingly similar to Punch Out.
Classic Game Room HD - MAZIN SAGA MUTANT FIGHTER review
Haha, that's a good one. I really don't think about this game much, even though I KNEW I had to have it, as I was a fan of "Tranzor Z" when it came out in the US in the 80s.