She was baptised Alexandrina after one of her godparents, Tsar Alexander I of Russia, and Victoria, after her mother. First official documents of her reign used both names but apparently she requested that Alexandrina be dropped
Apparently our Queen was asked which of her names she wished to choose as her regnal name and she was quite astonished. Read it somewhere.
Charles has quite a few to choose from and 'Charles' doesn't have a great record as a regnal name, so he may choose another - given his age, perhaps not, but it's up to him.