People already recommended some Final Fantasies, but really all of them can get you quite emotional at times.

If you like VNs, Umineko no Naku Koro ni made me cry so many times...

Edit: Just remembered Once on a Windswept Night, it's just a short game you can complete in a hour, but for some reason it made me really emotional.

Omori. If you don't cry when playing that game, you are a monster...

But more seriously, even though it's the best storytelling I've seen in ages, I wouldn't recommend it if you are depressed, that game hits hard.

Lisa: the Painful

I dont know if its listed here yet.

Yakuza Zero has a story that is excellent, albeit goofy at times. In the end, it got me good.

You don't mind porn? Go and play the AVN Acting Lessons. It's on Steam. It's not too long but the punch it packs... it has this reputation of "the AVN you'll never wanna play again". I like to compare it to Requiem for a Dream.
If you want less heart-rending tears, I just finished chapter 3 of Being a Dik and there were tears, but the kinda good emotional kind. I'm the first surprised, but it seems to be a signature of the author and I quite like it.

Otherwise I'm playing Metro Exodus and man, that game is oozing with that uniquely Russian despair. No tears yet, but I'm preparing for some dark moments and the whole thing is just... wow. Mad Max had beautiful everything, but M:E has fantastic writing throughout. It all feels deeply personal and with that dreadful feeling of inevitability. And you only get to see/hear it if you want, mostly. Very glad I gave this game a second chance (it's very... clunky. Again in that typical slavic way, I guess; but it works if you accept it as it is).

The Vanishing of Ethan Carter! I'm not going to say anything, except that you need to do all the "side" quests. If you miss any, the map at the end will show which you missed. See the ending!

Dear Esther is also quite special, the monologues change every playthrough and everyone interprets the game differently. Very impactful.

I can recommend all games from SIGONO's OPUS series https://store.steampowered.com/publisher/sigono/#browse

One I've not seen mentioned but made me bawl my eyes out is 'Presentable Liberty' Im not sure about the remake, but the original was pretty short and was gripping, despite the low level of action.

Final Fantasy X.

Probably gonna get hate for it but I'm a man in my 40's and some of the Genshin Impact story lines have left me in tears.

God of War: Ragnarok got me close to bawling. Father son moments always get me in movies and games, and this was one of the best.

Disco Elysium shattered me half a dozen times. Two moments in particular made me step away from my desk and go on a walk. It's the best-written game I've ever played.

Firewatch. Gorgeous, emotionally honest game that wrings you out.

Inside. More of a stressful, cathartic cry, but it applies.

Mass Effect 3 - but only if you play the first two. Incredible story that beats you over the head with yearning .

Red Dead Redemption 2 made me bawl my eyes out.

Rimworld - Randy Rough

Ghost Trick is a very twisty game, but a few of the twists had me tearing up.

If you've ever experienced grief, then Gris. I still listen to the beautiful soundtrack regularly