Is it a PNG with fake transparency or a different file format pretending to be PNG, or both?

It is difficult to tell from the context and I’d like to understand better.

I think it could be either, and I’ve seen examples of both in the wild.
I think the frustration is finding an image that purports to have a transparent background but doesn’t.
some sites uses jpg with fake transparent checkerboard image to be shown in search engine, forcing user to login or just go to their site to download the real png file