I'm just curious because Window Capture is mentioned in many an online forum and still there in the OBS Studio documentation, but in my version it's gone from the list of Sources.
I found an OBS Forum post where OP asks the same question; however, in their case, OP is running Mac OS, and the OBS kb mentions Window Capture's having been with superseded by macOS Screen Capture for Mac users.
I'm running on Bazzite Linux, Gnome, Wayland. OBS Studio version 31.0.3. I believe I'm using the flatpak version.
EDIT: Using Distrobox, I confirmed the Window Capture option has gone away at some point between versions 29 and 31 (at least for Linux users anyway)—full update w/ screenshots in the comments.
EDIT 2: After more tinkering, I figured out that the option for Window Capture hasn't gone away entirely, it's just been tucked away behind a couple clicks and renamed. To get to it:
With Screen Capture selected in Sources, click Properties (for me, there's a big Properties button above the Sources pane, below the Preview pane).
On the Properties dialog, click Open Selector.
On the next screen, there are two options: Application Window and Entire Screen—select Application Window.
There, it'll display a list of windows, one of which will be the game's window if the game is running.
Game Capture might be the same thing, I'd try that out
You might be right. The OBS kb says Game Capture is for Windows users only and that Linux users should use Window Capture. In my version, I have Game Capture, but not Window Capture. I fiddled with Game Capture, but after a few minutes I couldn't get it to show anything other than a black screen, so I went back to what I use now, Screen Capture (Pipewire).