If there's an ongoing narrative, naturally a sequel should continue that narrative.
Otherwise, I expect any sequel to build upon the foundations of its predecessor without losing sight of the game's core concept. I often point to Arkham City as a good example of what a sequel should be. It added quick-fire gadgets to Batman's arsenal in combat which expanded upon Asylum's already stellar combat without detracting from it. It expanded upon the predator stealth system the first game established while maintaining the games' core stealth play.
Arkham Knight tries but mostly fails to do similarly. Rocksteady tried to integrate the Batmobile into stealth and combat and puzzle solving but did so in a haphazard way. Though they tried to, they also couldn't really integrate the Batmobile into hand to hand combat or predator stealth in consistent or useful ways, so the Batmobile ends up having its own separate stealth combat and puzzle solving. Ultimately this ends up making the Batmobile feel tacked on and distracting rather than a fundamental addition to the core gameplay. The rest of the game is basically just MOOAAAHHR BIGAAAHR!! with a load of extra
stuff that just doesn't really add anything.
Arkham Origin (a garbage game you should never play) added a tether to Batman's arsenal which basically just allows you two free stealth predator kills for any encounter. It's more of a cheat than an essential addition to the gameplay. It doesn't do anything new. It even shallowly rips off gadgets from Arkham City. Instead of Arkham City's ice bomb, it's a glue bomb.
Zelda II is too different from the first LoZ game to make for a proper sequel. Of course the Zelda series is no stranger to straying too far from the series' core tenants, but instead of dumping on Wind Waker or BotW for the bajillionth time I'll find a new "victim." Zelda II adds some rudimentary RPG stat growth that has never returned to the series (though I do think it could be properly integrated) random combat encounters and a perspective change that changed the core nature of the experience too much. ALttP was the series' return to form, and an adequate sequel to the first game which built and improved upon the first game's foundations while remaining true to the series' core nature.
Riven, the aptly named Sequel to Myst, is also a good example of a proper sequel. It features more of the first game's striking visuals, world building, puzzle solving and narrative. Only the visuals are better, the world is better realized, the puzzles are better integrated into the world and trickier at that, and the narrative is richer. Anyone who has played the first Myst will feel right at home in Riven, only blown away by Riven's advancements.