Granted I live in southern CA, so not very cold, but it is very common to have horses clipped and blanketed in a light weight turnout at night. I have never boarded somewhere that had designated storage for boarders’ blankets. You got the blanket bar on the stall and your designated area in a tack room to fill as you see fit - there may or may not be room for a tote of out-of-season blankets, depending on how much other gear you had.
I keep mine in a sturdy plastic tote at home when out of season. In winter I keep one sheet and one blanket at the barn to swap out depending on the temperature. I have a spare blanket I swap out if the primary one needs to be washed or dried.