The social security data base is not automatically linked to state or county databases. For example, if a person is married and would like to apply for last name change, the person not only need to change the last in the State where s/he resides, the person needs to go to Social Security Administration to apply to have his/her name change. At that time, the person will have to show proof, document, and reason of name change.
Therefore, if you would like your last name changed or your step father's name added to your records, it is best that you go to SSA to ask for the proper procedure to do so.
Some of the procedures and explanations we find are on how a step parent can apply for a social security number for the child. See links below. It is apparently not very common but is crucial equally to find out how to also list the adopted parent's name on the record. Sometimes, the step parent may also qualify the adult children for benefit, etc. So, the SSA must have a way to add the record.
Fiona Chen, MPA, Ph.D., CPA, ABV, CFF, CITP