I apologize if this is very elementary: I've never heard of closet referred to as 壁橱， only as 衣柜， but the direct translation doesn't seem apropriate. I've also heard 衣柜 used to reference cupboard as well. Is this for colloquial reasons?
壁橱 can refer to a small closet, a wardrobe embedded in a wall, a cupboard hanging on a wall, a chest placed against a wall or something else. As the characters go (壁=wall, 橱=case), it can be a container for anything not limited to clothes. Note, as an imported concept, 'walk-in closet' is not translated into 壁橱 but something along the line of 储衣间 (clothes storage room) or 更衣室 (dressing room).
衣柜 (衣=clothes, 柜=cabinet) means a wardrobe, either built-in in the wall or a standalone piece of furniture. A general-purpose cupboard should never be called 衣柜.
In Chinese, the distinction between built-in in the wall vs. standalone furniture is not made by first-class words or affixes. Both 壁橱 and 衣柜 can refer to either a closet or something else.
For the similarities/differences between 橱 and 柜, please refer to @Question Overflow's answer here.
So, from there it's very clear. That a wall cupboard. In Mandarin, a word can remain very much of meaning. So, if only by a word, we can't know actually what they want to mean, after combine these words. We can actually know what it's mean. Actually if you uses these word to combine to other, it won't mean closet, but will mean other as what the sentences mean.
and 衣 = clothes 柜 = something like a place to store things. So, combine it can became CUPBOARD
I believe a
A similar kind of furniture with open compartments and no doors is known as 架 or a shelf.