Games like Tomb Raider

Haydee.

Excalibur 2555AD was a contemporary of the original Tomb Raider and plays similarly. It's on the PS1 and there was a Windows version but it's probably hard to get running on a modern computer. It's also not all that good, from what I've read about it.

https://en.wikipedia.org/wiki/Excalibur_2555_AD

Shadow Man is like old school Tomb Raider in a horror setting. There's a remaster coming out hopefully some time this year which will have more modernized controls.

you got a great list of games recommended here
but if i may add detective games like Sherlock Holmes
with more puzzle and less action
or if you want more action i hear good thing about immortals-fenyx-rising

also
https://steampeek.hu/?appid=750920
glhf

Middle Earth

If you are looking for a game about puzzles and about action and riddles, then I will recommend you the third person game Indiana Jones® and the Emperor's Tomb ™

A Plague Tale: Innocence
Ancestors: Humankind Odyssey
Alan Wake
Batman: Arkham series
Bloody Spell
Control
Dead Rising series
Enslaved: Odyssey to the West
Gujian3
Hellblade: Senua's Sacrifice
I Am Alive
Mass Effect: Andromeda
Middle-Earth series
Remember Me
Star Wars: Jedi Fallen Order
Thief
The Wind Road
Vampyr

Legacy of Kain: Soul Reaver 1, 2 and Defiance. I dunno, when I think of Tomb Raider I think of pushing giant blocks.

As many people already suggested, Enslaved and the Arkham Batman trilogy.

Batman Arkham series https://store.steampowered.com/search/?term=batman+arkham

Closest to Tomb Raider is Adams Venture Origins.

Read some games that people suggested, and I was like WTF, a lot of them have nothing to do with TR series nor their mechanics,

I can suggest you few more, BUT they are NOT entirely like Tomb Raider, but have few mechanics from Tomb Raider that might get your interest.

If you want STEALTH, then check out Splinter Cell series.

If you want NICE STORY, SKELETONS and a bit of PARKOUR, then check out Prince of Persia series.

If you want PUZZLES and a lot of THINKING, then you should check out Sherlock Holmes series, especially Sherlock Holmes vs Jack The Ripper.

I wont suggest Assassins Creed: Origins, it has few pyramids/tombs to discover, but 90% rest of the game is nothing like Tomb Raider.

Try Rime if you like the puzzle solving and platforming but it has no combat

I would say that you could like The Legend of Zelda: Breath of the Wild, but you would have to have a switch... (or Wii U)
I don't really remember much of the gameplay of Tomb Raider, so I can't say much about that, but:
3rd person - yes
adventure/action - yes, although the action isn't that intense (because obviously, there aren't any firearms)
puzzle/riddle - yes, there are tons of them! Most of the shrines have a puzzle, there are also many of them scattered around the world (although some of them may be too easy).

There is not really a point in buying a console just for a single game, but maybe in the future, when you get your hands on one, be sure to try it out!

Ever played the Legacy of Kain games?

Any Ubisoft game.

Urban Chaos, Bionic Comando 2010, Remember Me

Not sure if I saw it mentioned, but what about the Dishonored series?

The 3D Prince of Persia games feel like they owe a fare bit to Tomb Raider. Sands of Time is the best of the lot in my opinion, but they're all at least alright. Apart from Prince of Persia 3D, but that's not on steam!

Less of a platformer, but it hits the key points you mentioned in the post: Legacy of Kain: Soul Reaver. It's not a Tomb Raider clone by any means, but it's a 3d person action adventure game, with puzzles to solve and a pretty good story even.

Also, not on steam, but if you can track it down, Heretic 2 feels like it drew some inspiration from the older Tomb Raider titles. It's not a clone, but it's clearly a post-Tomb Raider 3rd person action/adventure game.

Indiana Jones and the Emperor's Tomb https://store.steampowered.com/app/560430/Indiana_Jones_and_the_Emperors_Tomb/ old, but great

Prince of Persia. Uncharted series. Assassin Creed (Though its open world, so not the same?)
Devil May Cry is more of an action third person game. Same for Bayonetta.
Currently I am playing Control, which is amazing.

And finally, you can always play the older Tomb Raiders, which I felt were better than the new ones.