They
might not be King Arthur. But you're right -- they get to choose which name they will rule under, so however unlikely, they
could choose to be King Arthur. (But would Charles be Arthur I or Arthur II?)
Charles may well choose to reign as Charles III since he is so well known by that name already, and who knows when he will ascend the throne. On the other hand, his grandfather reigned as George VI, even though everyone had known him as Prince Albert (or "Bertie"). In fact, three of the the last six monarchs ruled under names other than the names they were typically called.
Some have speculated that Charles may not be inclined to be Charles III, since history has not been kind to King Charleses in Britain. If he is not King Charles, he will probably be King George VII. Very unlikely that he would be King Arthur, given the legends surrounding the idea of the return of Arthur and the baggage using that name would bring about.
So back to main topic, the Loch Ness Monster is on a list (#56) of "The 101 Most Influential
People Who Never Lived"? My son would take issue, both with the Loch Ness Monster being a person and with the contention that it never lived. He's sure it's there -- it's one of his favorite things to read about. He's currently writing a story about it in 3rd grade. (No. Not proud at all, not even while I type this and look at the Nessie on my desk.

)