Also, there has never in history been as quick and wide a rollout of a vaccine as COVID-19. If a database were tracking every negative outcome for a person after taking a vaccine - even if the vaccine didn't cause it - we would see a spike. You'd see a worrying spike in negative outcomes if we spun up a database that recorded adverse reactions after eating a banana, for example; doesn't mean bananas are killing people.
[1] https://jamanetwork.com/journals/jamanetworkopen/fullarticle...
[2] https://arstechnica.com/science/2022/01/up-to-76-of-covid-va...
Putting aside the obvious difference in societal impact and media coverage, which the modern flu has never received, it seems like the drastic differences in vaccine uptake absolutely contributes to this.
Flu vaccine uptake in 18+ was estimated at 45.3% in 2018-2019, 37.1% in 2017–18, and 43.3% in 2016-2017 [1]. It's sitting at 76% for COVID right now [2].
1: https://www.cdc.gov/flu/fluvaxview/coverage-1819estimates.ht... 2: https://usafacts.org/visualizations/covid-vaccine-tracker-st...