I've been jabbed a lot for trips to foreignland, so a flu jab was nothing. I used to get one despite being a healthy, youngish individual because the mothership jabbed us. That said, the last time I had one a couple of years ago, I got quite ill afterwards, so haven't done so since.
Logically, I know the flu-like symptoms afterwards were a coincidence, there's no live virus in the jab. But the monkey part of my brain didn't like that explanation.
The same social distancing and other preventative measures act against the transmission of influenza as much as COVID, of course. That said, the only thing that stops you getting flu and COVID at the same time is probability.