There's something wrong with that website. I have never heard of any such expression. WAIT. Correction. It's something in the Harry Potter books. It isn't A "hat stool," it is THE "SORTING Hat Stool," and it is THE stool on which THE "Shorting Hat" rests on during the Sorting Ceremony. It has no meaning outside the Harry Potter universe. It is crazy for a "word learning" website would include it.

I just had an unpleasant thought. "At stool." "Stool" can mean "toilet" (or worse yet, the things that go in a toilet, as when a doctor wants a "stool specimen"). There is a COMPLETELY OUTDATED phrase I've read in old books. Where we would now say "using the bathroom," in the old days someone was said to be "at stool." It's sometimes still used that way in medical books.

Maybe it was a typo? I have never heard of 'hat stool'. We sometimes name things according to the context they are used in - for example, we have a boot box, a shoe place, a comfort blanket, so maybe in context it would make more sense?