"Chest of drawers" is its name. It is quite a long name for a common item, but that is what it is called. It has been around for a long time, the expression has been in use since 1670 (etymonline)
Similar items include "tallboy" (a tall chest with drawers in a column), A dresser (usually a bit more than just a chest of drawers, with a top part for display). A bureau (a combination of drawers and a writing desk). But there is no word that means exactly the same as "chest of drawers".
In many situations you don't need the full name.
Put your socks in the drawer!
There are similar examples of other things named this way when the item is a "container" of some sort.
A pack of cards (not a cards pack). A bunch of flowers (not a flowers bunch). A bottle of beer ((when talking about the beer, but beer bottle is also fine when refering to the bottle).
(ngrams searches for card pack and pack of cards "Card pack tends to refer to specialised packs "The game contains an 8 sided die and a card pack showing countries of the world..."
drawer chest and chest of drawer "drawer chest" nearly always phrases such as "a five-drawer chest".
And remember all the collective nouns:
A herd of cows. A flock of sheep. A pride of Lions etc.