Steam Overlay enables you to do other things while gaming, such as browsing the Internet, usually for guides, or chatting with friends. And if you're the kind who likes to take screenshots, the overlay must be enabled for you to be able to post them.

Sadly, for reasons unknown some games just don't have Steam Overlay. Other than this, I've experienced it too in Thief: Deadly Shadows, Tomb Raider: Legend, and Tomb Raider: Underworld. Following this formula, you should be able to enable overlay in games that don't support it and get some nice screenshots.

First of all. open the Run application (Win+R) Then, copy and paste the text below. The formula to enable overlay is:

overlayuipath -pid gamepid -manuallyclearframes 0

But, it's obviously not finished. You have to change 2 parts of the formula to make it work. You must do a little investigating.

Usually, it is located in your Steam installation folder. Usually, it's in:

C:\Program Files\Steam\GameOverlayUI.exe
or
C:\Program Files (x86)\Steam\GameOverlayUI.exe

When you've located it, hold SHIFT, press right click, and click on the "copy as path" option. Or if it's in the same location, copy it from the text above.

Next you will have to find the Process ID, or PID. To do it, you must first run the game. Then, Alt+Tab to change to your previous, non-game window and open the Task Manager (Ctrl+Shift+Esc). Open the Processes tab, and change the PID in your Run window to match the one on Task Manager.

IMPORTANT: Remember that the PID changes every time you close and reopen the game, so you must do this each time.

So basically in your Run window the formula at first was:
overlayuipath -pid gamepid -manuallyclearframes 0

and now it should be:
C:\Program Files\Steam\GameOverlayUI.exe -pid 7268 -manuallyclearframes 0

Press Enter on your Run window and go back to the game.

When you go back to the game, wait a few seconds, and a familiar notification will appear on the edge of your screen:

Now you can take screenshots. But you cannot use the web browser since it won't load, neither can the Friends tab. You cannot browse or chat, but you still can take screenshots.

Credits to Melody ♡ for discovering this method which helped me a year ago. Link to the guide: https://psteamcommunity.yuanyoumao.com/sharedfiles/filedetails/?id=1138005386
I assume it works differently on different OS such as Mac or Linux. If you know how to do it on other OS and want to make a guide about it, feel free to let me know and I will link yours here.