Because the dynasty names are often more like adjectives to describe an era, even though the names were chosen by the ruling family.
Ming means "brilliant", Qing means "pure, clear".
Also, having a given Chines surname doesn't make you very distinctive.
There are only about 100 surnames to go around. So whatever your surname is, you can be sure that there are people from all walks of life with the same surname.
The surname of the ruling family during the celebrated 300-year Tang dynasty was . . Lee.
Know any Chinese people whose last name is Lee?