If a cheese sold as pasta-cheese does not melt, is it "cheese"?
That's a processed cheese product. It may be allowed to be called cheese in some places, but that's a matter of product labelling regulations where you happen to be.

The melting is a different issue. I'd expect that product to melt, though possibly not if it has the texture of parmesan - parmesan doesn't really melt on its own though it melts into sauces. And parmesan is commonly served on pasta, so a non-melting (or not really melting) pasta cheese is a thing.