I agree that, at least for AE, "school supplies" is what would normally be said in this context, and when I hear/see the term "stationery" on its own, I would normally associate it with the paper-type products that have been mentioned, but M-W thinks differently:
Definition of stationery
1 : materials (such as paper, pens, and ink) for writing or typing
2 : letter paper usually accompanied with matching envelopes
If I were to visit a stationery store or a stationery department, I wouldn't be surprised to find the items in #1 there in addition to those in #2, but, as I have said, used on its own, I would tend to understand it in the meaning of #2.
AHD basically offers the same definitions as M-W, but in reverse order (which I find more appropriate).