In Hungarian it's the same with "home" in particular. You say "I'm home.". In Hungarian, I too say the exact same thing: "Otthon vagyok" (I'm home).
Your other two example works the same, you won't say in Hungarian "I'm school" (Iskola vagyok (it means I am literally a school)). But you say "IskoláBAN vagyok" (I'm at school) or "PostÁN vagyok" (I'm at the post office. Notice the suffix in this case is completely different, but that's another story of Hungarian)
okay, so this means the word 'home' is actually special accross languages 😆.
and not neccessairly the home as homeland like haza in hungarian ('cause that's not even a noun (tho it is somewhat equivalent with home)), home like... your home.