I play a lot of Fate/Grand Order, a Japanese role playing game with gacha mechanics based on the Fate franchise. Other than that, I'm fairly particular in the games I play. I don't think a lot of games on mobile are particularly well suited for touch controls. The exceptions include management, music rhythm games and shoot 'em ups. So I've played many Kairosoft games, which to me are as compelling as any of the Maxis games back in the day. And I've played a fair bit of Groove Coaster, Aka To Blue, and the Danmaku Unlimited games. They're just all very well suited to the platform.
As far as emulation, I only really like using my phone for Nintendo DS emulation, but I'm interested in seeing how Citra for mobile develops. I've also used VNDS to port some games in the past, including the Type Moon games.