Yes, people use the expression to refer to any small, inexpensive Christmas gift. The term originated with gifts that could be put in the stockings that children hang up "by the chimney with care in hopes that St Nicholas soon would be there," but the concept has been expanded to include gifts for adults and does not require that the item actually go in a "Santa stocking."
You're right that the poem doesn't say whose stockings were hung, etc. In my childhood home, only the children had stockings, so I have been laboring under a false impression for half a century. Thanks for the correction.I don't think the classic and iconic poem, "A Visit from St. Nicholas," limits the stockings to the children of the household, and when I was growing up the adults as well as the children had "stockings" to hang on the fireplace.