Florida law? Definitely not.
The UCC, including Florida's version, says a customer's signature is whatever he wants it to be, so I don't think there are going to be any laws telling the customer what his name has to be.
Your bank can reasonably have a policy that it will only open an account in the name for which the individual has identification. Regardless, if the customer's Florida drivers license says he is Nathan Bedford Forrest he should be able to open an account:
Nathan Bedford Forrest, or
Nathan "Wizard" Forrest, or
Nathan Forrest, or
N.B. Forrest, or
N. Bedford Forrest
No nicknames or tribal names unless they are reflected in the ID or the account title also reflects the name on the ID. (Keep in mind, a customer can order personalized checks online that say anything he or she wants them to say.)
The only place where the concept of a "legal name" actually has meaning in on real estate transfers. There, it's about consistency and satisfying the lawyer, not a law or regulation.
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In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.