If someone is allergic to benzocaine could they be allergic to all chemicals that starts with benz?

Not likely: Allergic reactions to local anesthetics such as Benzocaine are a rare but known clinical phenomenon but don't indicate allergies to other classes of medications with benz- prefix in them.

Benzocaine: Benzocaine is a local anesthetic commonly used as a topical pain reliever. Just because you have allergy to benzocaine, it doesn't mean you have allergy to other local anesthetics.