Common cold is caused by a virus. And yes, it mutates quickly. Several strains are usually in circulation at any given time, and they will change by the next flu season.
My understanding that there have always been thousands of different “cold” viruses, so one cannot be immune to all of them. They just all cause similar reactions so are called the same thing generically. Same with influenza. But the chance of one suffering from the very same species (if you will) is almost nil after having had it.