C-store bride officer unlocks can only be claimed once. So, if you dismiss your Borg bridge officer, you won't get another for free, it says this in the item description:
If you discard, sell, trade, or use a Borg Bridge Officer for training, you WILL NOT be able to claim a free replacement from the C-Store. You can only have a single Borg Bridge officer active at once.
A while ago I got a Borg officer and obviously I only get one so I can't have multiples, either on the one character or others.
If I dismiss her, do I lose her forever or can I hire another Borg character on another character?
When did you buy the borg bo? If you purchased it before they put in effect the f2p c-store, it will be an account wide unlock look under the legacy tab like others have said. However, I am not sure but you may only be able to claim it once per character.