As a speaker of British English I am not at all sure what a 'closet' is, let alone a 'bottom closet'. (I know what a water closet is, and what is meant by coming out of the closet, but that's it)
Is this what I would call the 'bottom drawer' of a 'chest of drawers' ?
Or are there two 'cupboards' here, one on top of the other ?
It's a cupboard. Yes, two cupboards one on top of the other, or one cupboard divided mid-height with two separate doors.
As a speaker of American English, based on your description, I'd say that the blankets are in "the lower cabinet." (I found "the bottom closet" weird too.)