The way that this Covid vaccine works is quite different from the flu shot. The Covid vax. is not a (dead) bacteria or virus being injected into your body. (With Pfizer & Moderna) RNA (called messenger RNA) that's been coded in a lab goes into your arm. The RNA has instructions, or a code, that gets deposited on the cell. The ribosomes of the cell make a spike protein. It looks similar to the spike protein on the surface of the actual Corona virus. Then your body is supposed to make antibodies (antigens) that goes after the spike protein and neutralize it.
The problem is that this has not been tested in any long-term studies, so, this is more of an experiment. We don't know whether or not the DNA gets permanently changed (possibly bad). We're talking about something that's not supposed to be in your body (made in a lab) that can bind to the surface of your cells and possibly may not be able to be reversed.
Also, since the drug is still being evaluated for long term adverse effects, pregnancy, etc., the FDA would not license the drug, but instead, put out as EUA... Emergency Use Authorization.