Flu shot reaction question

First of all, the issue is resolving, but I’m curious. I’m curious because I have consulted two doctors, a dermatologist that I work for, and my own primary care Dr. So not really looking for medical advice, per say, but opinions.

On Friday at noon I got a flu shot at work. Within a few hours the site was red, swollen, hot, and intensely itchy. I mean intensely, not just a little bit. I spent all weekend scratching it like mad, and now on Monday at 5:00 it’s subsided a lot, maybe due to the Novacort cream I got from my employer, and appears to be going away.
Neither of the doctors I asked about it could tell me what kind of reaction this is. Apparently not a systemic one as I have no other symptoms but local. Google shows nothing about flu shot site reactions and itching. I did wear a bandaid for a couple of hours and have sensitive skin. I wondered if the MA that gave it somehow missed the muscle, as I have none of the typical soreness of the deltoid, and the fluid became trapped right under my skin. But then I thought, no, I just have an overactive imagination- my deltoid is well-developed and I don’t have saggy skin.

Does anybody have any suggestions as to whether this means that I am allergic to the flu shot, and should not have it again?

I get a bad reaction every year, very swollen, painful and itchy, and it lasts several days. Other people get no reaction at all. I’ve heard that if you get a bad reaction, that means you “need” the shot, but that makes no sense to me, whatsoever (how could you “need” it before you’ve been exposed to the virus?). So I assume it’s some sort of allergic reaction.

I am not a doctor, and have admitted to being an anti-social killer, but it could maybe just possibly be an allergic reaction to eggs, which I hear is used to make the stuff. An allergest could answer for sure.

99% of the population find these shots to be immensely beneficial. You, unfortunately, are part of the remainder. Them’s the breaks.