It could be the flu it could be covid could be both could be other things going on.
I did read that the flu shot this year was a poor match with the dominant strain.
It's more that the doctor, even if it wasn't your own, assumed it's the flu so no big deal, that's no good. Plus it's not good advice to say you have lots of protection. At least with Pfizer there is a decrease in protection month after month in the limited studies done. I can't remember the stats with Moderna. Plus the doctor wouldn't be able to say exactly how much protection your particular body generated.
Omicron is presenting different symptoms than Delta variant if it's covid. Omicron is more cold-like, Delta more like the other strains with the other symptoms we're used to hearing so just keep that in mind.
Be aware with the at-home tests that because they are generally rapid tests they have a higher rate of inaccuracy. Omicron from recent information is presenting more issues with accuracy of at-home tests because of the different protein spikes. Do not rely on at home tests solely.
As far as breakthrough cases yes they occur. You're actually almost to the point where you should be getting a booster (strongly recommended once you past 6 months after your 2nd dose if it's Pfizer or Moderna or 2 months past your 1st dose with J&J).
I wish you well and that you can get a test and it is not covid
FWIW my sister-in-law went in for covid testing early last week because she felt off. It was not covid but a sinus infection and they offhandedly asked if she wanted a strep test and she said sure. The strep test came back positive so she had both a sinus infection and strep. Hard to remember other stuff still exists but it does!