Answer
Do screenshots contain GPS location data?
No. A screenshot is captured from your display, not from your camera, so there is no location fix to record. It does still carry a timestamp and some device information.
Updated
GPS coordinates get into a photo because the camera app asks the operating system for a location fix at the moment the shutter fires, and writes the result into the file. A screenshot involves no camera and no location request, so there is nothing to write.
This is the one area where the common advice is straightforwardly correct: if you screenshot something, you are not leaking your coordinates through the file's metadata.
What a screenshot does carry is a creation timestamp, frequently the device or operating system that produced it, and a colour profile describing your display. On some platforms the device model is recorded too. None of that is your location, but a timestamp plus a device model is not nothing, and it is worth knowing it is in there rather than assuming the file is blank.
The larger risk with screenshots was never metadata. It is the contents of the frame: notification banners, an open tab, a visible filename, a username in a corner. Metadata removal does not touch anything you can see, and on a screenshot what you can see is usually the whole problem.
You can check any screenshot yourself with the EXIF viewer — it reads the file in your browser and shows every field it finds.
More on this: GPS location remover, Metadata remover.
Terms used here: EXIF.