Are fatigue and dizziness considered allergic reactions?

No: While fatigue and dizziness might happen together with allergies, the allergy symptoms (itching, nasal congestion, sneezing, wheezing or vomiting, depending on the type of allergy) would be very prominent. Fatigue and dizziness alone are unlikely to be due to allergy.

It's Possible: But many things can cause this. Unless you had an allergy shot just when the symptoms started, you really should see your doctor for a better answer and evaluation.