Can you be allergic to another person?

Yes. Women can develop allergy to their partner's semen leading to anaphylaxis during intercourse. Fortunately, this is a very rare problem.
No. No, allergies are to specific proteins. More likely, you're allergic to something that person wears like a natural fiber of clothing, a perfume/cologne, a lotion/cream, or something external on their body or clothes like cat dander for example.