Neither Captain names or Ship names are unique as far as I can tell. Just your account handle (which is why for example I'm Jinni@Keo I'm pretty sure you could go register Jinni if you wanted as a Captain name).
Cryptic was smart in doing that. This way someone tries to spam you in game with credit selling site you can report them and ban their whole account. It won't be like other games where they can simply delete their char before you catch on.