Changing a Child's Name: The Birth Certificate and Social Security Card Last Name Does Not Match

My question involves name change laws in the State of: Washington

My husband just found out that his daughter has his last name on the birth certificate but his ex got the kid social security number with her last name.
She states that does not matter and that the kid won't have any issues to get passport or anything else, is just like if she gets marry and changes her last name.

Please I just want to know if that is correct or does affect the kid or my husband in any way.

Changing a Child's Name: The Birth Certificate and Social Security Card Last Name Does Not Match