The Legend of Nayuta: Boundless Trails has in-game achievements accessible through a trophy icon in the game's system menu.

Peglin has also included in-game achievements

XIII - Remake has in-game achievements.

Triple Town has in-game achievements.

Thank you all for the contribution. Sorry for the delay but the list should now be up to date again :)

Days Gone has them. They're called trophies in the Sony lingo.

Sword of the Necromancer has in-game achievements.

Midnight Fight Express has in-game achievements (can't be found in the main menu, but in the in-game hideout menu instead)

The first one was already on the list but I added the second one. Thank you.

Caves of Qud now includes in-game achievements.

Have any Spiderweb Software games been mentioned? I know that at least Queen's Wish 1 and Geneforge 1 - Mutagen have in-game achievements, which it calls "Medals". (I'm guessing that every sufficiently recent Spiderweb Software game does this.)

BallisticNG has in-game achievements.

Did a search for Tinykin but didn't see any mention of it in the thread.

It does have in-game achievements though. See screencap below

Edit: Not sure why the attachment is missing. Here's another link: https://imgur.com/a/9xXqbuA

Thank you both, added them to the list.

Thank you. For now I have just added the games which you have confirmed ingame achievements for. I will add other spiderweb games to the list after it has been confirmed that they also have ingame achievements.

Normally you you should not have to search through this thread since I am trying to add everything to the list in the OP asap ... sometimes I am a bit lazy and delayed but in that case it still should be enough to read the last page :)
Thank you for the addition which I have added to the list.