I think I have the same understanding of "non-fiction". But then I see poetry in the non-fiction section and think, that's not non-fiction, so it has to be fiction. Or more generally, non-fiction is based on facts and aims to impart those facts, to inform and educate. Everything else is fiction; it might also be fact-based but it's aim is different and it isn't purely facts. Possibly a different category error. Possibly the library understanding is fiction = stuff which tells a story, non-fiction = other stuff.
Clare says poetry is classified with literature, which further muddies the waters for me. What then is literature? Anything written can be literature, but unless specified it implies to me primarily fiction. Clearly writing about literature is non-fiction – literary criticism, textual analysis and so on – but literature itself? Often of course the two are in the same volume.
Anyway, the fiction shelves of libraries are not stocked alphabetically by author in one straight series; they have subdivisions, such as sci-fi, crime, graphic novels, short stories. I'm not a great fan of minute classification but poems and plays seem to me to be obvious and valid divisions and that's where I would put them. But the main thing I guess is that they are in the library and we can find them somehow.