Is King Arthur fake

Probably, but nobody really knows for sure.

All the historical references to him are either clearly fictional/fake or were written long after (like, hundreds of years after) he was supposedly king, and his actual existence is considered dubious. The general position of historians is that there MAY have been a "king", or at least local chieftain, whose name was at least somewhat similar to Arthur, but that if so, we don't really know anything about him.

The various Arthur legends are pretty clearly just that: legends.