Both “off” and “out” are translated to „aus“ in German. „Ausziehen” could be translated as “to pull off” (for clothes) or ”to pull out” (for houses and apartments). Hence the two meanings.

When you put your clothes on or move in you don‘t have the same problem. “On” and “in” are translated to „an“ and „ein“, respectively. That’s why there is „anziehen“ (“to pull on”, for clothes) and „einziehen“ (“to pull in”, for houses and apartments).

(To the grandparent: You really should try you luck creating compound words in German even when you are not sure whether the word exists. That’s a very German thing to do :)