Answer:
Discouraging it is fine, but prohibiting may be an unenforceable request. The employee who says "I work at ABC Bank" hasn't really done anything wrong. If the employee's use of the bank name implied an endorsement of some kind to a product, service or website, that would be another matter.
You may also want to discourage it from a security standpoint. Identifying one's employer may open a bank employee who is a Facebook poster to needless risk. Once identified, he or she could be targeted by a criminal as an asset of the bank.
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