Comparisons to the flu and colds aren't scientifically valid. Influenza mutates much faster than SARS-CoV-2, which is why the flu vaccine is reformulated every year. Common colds are caused by dozens of different viruses (including at least 4 other coronaviruses) and gaining immunity to one of them won't protect you from the others. SARS-CoV-2 also mutates but at a much slower rate than influenza.

Reinfection by SARS-CoV-2 is possible but rare. There may be some benefit to vaccination even for those who have recovered from infection, although the duration of additional protection is unclear.

https://www.medrxiv.org/content/10.1101/2021.08.24.21262415v...

https://www.cdc.gov/mmwr/volumes/70/wr/mm7032e1.htm