We have coronaviruses. There is no "cold" virus really and it's an umbrella term for a few different viruses including coronaviruses. With human adapted coronaviruses - unlike zoonotic covid-19 - you don't get lifetime immunity https://www.ncbi.nlm.nih.gov/books/NBK7782/

You can just think of booster immunizations and getting tetanus shots when you're exposed. Why would you need those, otherwise? If you want to learn more looking into memory T and B cells.