Just because a common cold is of the same family of viruses doesn't mean that it causes the same symptoms and has the same long-term effect on the immune system.

We don't know what the percentage of Omicron patients will be who develop long COVID. But there is no such thing as "long common cold." It's a much milder infection and doesn't seem to cause post-viral syndrome, as COVID does.

I agree with dfwl28, we're going to be seeing long COVID cases from Omicron. Studies from previous variants have shown that MANY people who have very mild cases do get long COVID symptoms.