If mom remarried in1960s..without legal name changes. USA citizens must use birth certificate name ID to comply with the laws ?

not true US citizens must use birth certificate name.

None of what you say makes sense. She doesn't have an obligation to change her name when she gets married.

Yes, I agree. The answer is certain.

A lot of women didn't change their name when they married. So what? A birth certificate isn't ID - it is proof of age and citizenship. ID is a document proving the birth certificate refers to them; e.g. a driver's license, school records, marriage records.

If she is using her married name then she doesn't need a "legal" name change - she just starts using it. If she has a driver's license she will need a certified copy of her marriage certificate. She needs to change her name with social security - she needs to show her marriage license which shows her former name and current name.

Do you actually have a question?

Ask someone who actually speaks English to help you compose a question that makes sense.