It might not fit entirely in, but check out Spec Ops: The Line. It's more or less linear shooter, but the story (and ending) makes it quite unique and very different from any other game I've played. Could too get pretty emotional at times.
Patient Gamers
A gaming community free from the hype and oversaturation of current releases, catering to gamers who wait at least 12 months after release to play a game. Whether it's price, waiting for bugs/issues to be patched, DLC to be released, don't meet the system requirements, or just haven't had the time to keep up with the latest releases.
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Try SIGNALIS ! It is extremely depressive, emotional and evocative, and to an extent it is a homage to stuff like Silent Hill 2
Ori and the Blind Forest? I know it's not quite as mature as some of the things suggested here, but it really knows how to hit you hard at some points throughout the story.
This may be from a different angle, but here are two games about grief:
Spiritfarer
That Dragon: Cancer
Reply All did a truly heart-wrenching episode on the second one.
came here to say spirit farer. that game is responsible for normalizing me hugging my spouse for no reason besides it feels good. Also definitely made me cry more than pretty much any other work of fiction in any medium.
Don't think I've seen anyone mention Oneshot. Amazing little game.
Plague Tale (Both games are solid, but 2 was sooooo good)
Had a quick scroll through and didn't see it anywhere yet, so:
Dear Esther.
Actually, reminds me I should play through it again, it's been a few years.
Persona 4 Golden is a beautiful game that made me emotional more than once.
Papers, Please. Maybe not crying, but could definitely put you into a depressed mood if you manage to get into character. Needing to make a choice between letting some poor bastard's wife follow him without proper papers and feeding your family is tough.
Cyberpunk 2077 and Horizon Zero Dawn are both fantastic games with an emotional storyline.
Haven't actually played it yet but a video about MOTHER 3 made me cry just talking about the story (not only that but the part where he talked about the story made me a little emotional)
Youay already know it but Omori was heavily inspired by the mother series and especially Mother 3: https://twitter.com/_omocat/status/1384687971929006080
Yakuza 0. That ending. Bruh.
Death Stranding
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.
The Yakuza series beyond Zero is also bloody amazing.
One moment you're laughing your ass off and the next thing you know you're in emotional anguish. If you're okay with Japanese weirdness (oh trust me it gets really weird), this series is a great rabbit hole to get into
The Last Day of June is quite lovely.