Strange question of the day: What would be the ramifications of identical twins growing up with the wrong names?

A couple has identical twins. They get the birth certificates and SS numbers for them. But early on they mix up the names so the kids grow up using the wrong names but nobody realizes it until they’re grown.

What kind of legal ramifications would there be?

Don’t get too serious on me! I don’t know how they’d ever find out they were using the wrong names. Just go with it!