This app was mentioned in 4 comments, with an average of 1.00 upvote
Well, I've been playing Mousehunt for 11 years. It started on Facebook but now can be played without a Facebook account with no real loss of functionality. I do suggest creating an account on www.mousehuntgame.com and linking it to the mobile one because it's very much a browser-first game. But the app certainly keeps pace and all the newer content is developed with both platforms in mind.
I have no idea what genre it would be in but the weirdest part of gameplay is there's a horn you can sound every 15 minutes (timered game?) but "friends" in the same area can also take you hunting as long as you've been active in the past hour.
It can be f2p but even if you do pay for stuff it's not really competitive so not p2w, more like pay to advance faster.
https://play.google.com/store/apps/details?id=com.hitgrab.android.mousehunt
Go there and check if there's an update waiting. You should have version 1.68.3 by now.
Currently playing Candy Box again. It's a short game, so don't expect to be playing it for weeks or months.
Another excellent "passive" game, though totally different from TT, is [MouseHunt] (https://en.wikipedia.org/wiki/MouseHunt_(game\)). You can download the app and play purely on a mobile device (iOS or Android), but I'd strongly recommend making the account via the dedicated site (which requires a Facebook account) or FB itself, due to all the other features.