I lost my birth certificate when I contacted New York to get another one I sent all my information and when they contacted me they stated they didn't have any records of my birth then they called back and said they found someone with my date of birth and first name but the last name was different I went to get my birth certificate when I was 18 and a man claiming to be my family started to send child support when I was 20yrs old and he changed my last name to his is that legal if the person is an adult and I didn't give consent and my mother didn't either

Both parents have to give written consent in NY.

I am confused - you went to get a because when you were 18. "A man" then started to send you child support. Two years later that same man changed your last name on your because.

Do I have it right?

It's a Supreme Court procedure in NY. You need to find out when/where in NY this was changed and take a look at the file.

The title of this question says a name was ADDED to your because. Are you saying a name was ADDED or your last name was CHANGED by this person?