How to make the player log into itch.io in my game to confirm that he bought the game on Itchio?

itch.io is DRM-free by design, so I doubt this is technically possible.

There is a long thread about copy protection. The summary is that it doesn't work, hackers can bypass it in hours and release a cracked version, numerous studies have shown that DRM decreases sales and revenue, and only really inconveniences paying customers. So I would recommend just forgetting about it.