Hoarding is a mental illness and it sounds like your mom brought it into the nursing home with her, which isn't surprising. If the aid comes daily, can she "recycle" i.e. take the used tissues and bottles/cans with her every day? Anti-depressants sometimes help those with hoarding to be less compulsive about it.
If there's any reasoning with your mom, you could also tell her she's in danger of being kicked out of the place (if that's true).
Thanks, everyone, for responding! She does not have dementia or Alzheimer's; she is simply stubborn and almost compulsively set in her ways. She is losing her hearing and her eyesight and refuses to be examined to see if anything can be done. Her bed is FILLED with everything from clothes to adult pullups to tissues (used) to bags filled with empty drink cans and bottles that she saves for the aide to recycle. She cannot throw anything away and cannot (literally, she sees no problem here) understand why anyone else would throw anything away. I am worried sick that she will get kicked out of the nursing home!
Does she have dementia? If so, you are not going to "teach" her anything. Someone will have to either prevent her from doing it or undo it frequently before it gets overwhelming. The clear plastic boxes sounds good. She may come to like it better than the bed storage.
Maybe your Mom needs to "see where everything is". Ask the NH if you can put a open bookcase in your Mom's room instead of a dresser. Then put some clear plastic containers on the bookcase. Your Mom might just end up putting the plastic containers on her bed... but if she keeps her belonging in the containers, that is less mess and a "WIN" for you. :)
What kind of "everything" is she putting there? The simplest solution is to reduce the amount of things she has, if it is mostly clothes then perhaps the NH would store most of them in the laundry area?
If there's any reasoning with your mom, you could also tell her she's in danger of being kicked out of the place (if that's true).