Not just medieval. that practice continued in the 1800s, up until photography. The rich would hire a painter to do everything, but those with less wealth - or the more frugal rich - would hire a painter to just do the face and hands. Hands are important, not just the face. Everything else about the body is covered by clothing (notice long sleeves - avoids having to paint the arms), and so can be done in a more convenient location. I wouldn't be surprised if the practice predates writing as is is really obvious and some people have more a talent for painting.

In Iowa painters would spend all winter (when snow made it hard to get around) painting bodies. then in summer when getting around was easy they would travel around, whenever someone hired them for a picture they would select a pre-painted portrait and add the face and hands. In the Iowa governors mansion there is such a picture where the guides point out the the boy in some picture was holding a club (probably for croquette) in an impossible way because the club was already in the picture. (the picture is from the second owner of the mansion, long before the state bought it)

It is in use also today in turists destinations. Put your face through a hole in a wall painted with something (body of an animal for example), get a picture.

Do you have any sources where I can read more about this practice?

All the way back to the Romans, the statues had detachable heads so that one ruler or governor's head could be replaced by the next.

The actual governors had detachable heads, too.

On along enough timeline everyone has a detachable head.

Did they care about different body types? Or could you end up with Trump's head on Obama's torso?

I've seen a few of those paintings. Some were of children's faces on grown-up bodies.

I mean have you seen some of the Trump fanart?