It can be, yes. A closet built into the wall is called a 벽장 but no one is that pedantic.

It refers to the container. The 欌 (장) in 옷장 is the same as in 책장, bookcase/bookshelf

Definitely lookup wardrobe on wiktionary to get a better idea. It's an archaic/bougey distinction that can be made against closet but most people today would regard them as synonymous. Wardrobes are theoretically clothing cabinets that aren't part of the floor plan like a closet. My grandparents had one and they are nice in a studio apartment if you can afford to buy one. I just had a wheeled hanger bar apparatus with no enclosure when I was in a studio. Synonyms for wardrobe (movable furniture for storing clothes): armoir, dresser; cupboard (UK); closet (regional US), press (Irish & Scots), shrank