If you will be feeding it to someone who has a FISH allergy, the answer would be "No". While a fish allergy is not as common as a shellfish allergy, it would be better err on the safe side and either cook the chicken and fish separately with fresh oil; or cook the chicken first and then the fish stick. The latter is probably what you would be doing anyway if the fish allergy weren't even an issue because you don't want to the chicken to smell fishy.
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You cannot reuse the oil for a second batch of chicken without risking an allergic reaction to fish.
Another option is to have separate fryers, one reserved for chicken, the other reserved for fish.
Best of luck.