Did you try to run the purchased game via the itch app or directly?
What you describe is DRM and that is quite rare on itch. I saw it maybe twice. And to my knowledge there is no support for DRM within the itch app, but maybe I am mistaken. (Meaning it should not matter, from where you run the app)
So basically, complain to the dev, they probably screwed up their DRM scheme, once again proving, that it is legit customers that suffer from that concept.
A likely scenario would be, that the app implements it's drm by phoning home and the server that indie dev put up could not be contacted and your purchase could not be verified.