The requirement of compliance with the license in the absence of documents on the right to possession of a software product

Greetings to you, dear habrasoobschestvo! One question came to me ... And he began to worry me more and more. It came to the point that this post was born ...
The fact is that I put the site engine found on the Internet. Carefully shoveled it, removed the copyright. The site lived and developed for a long time. But the “owner” of the engine has recently appeared, which requires you to put copyrights on the site, threatening everyone, from abuzy to hoster to personal reprisal :)
I believe that the engine was written by this person, and I am ready to put this unfortunate copyright. But the question is - does he have the legal right to demand to leave copyright? Naturally, the author does not have any written documents on his software product. And in general, does he have the right to demand compliance with the license if his right to the engine is not documented? I ask the community to drop the moral part of this issue and share the experience.