Anonymity is a special case of privacy. Privacy is the ability to control how much you reveal about yourself to others and the circumstances of how you reveal that information. Anonymity is simply the choice to reveal no information about your identity.
To some extent they are the same, and to other extents they are not. That is why they are two different words, with some conceptual overlap. Anonymity is a type of privacy, and privacy (in the US at least) is a concept that appears in the US constitution and in subsequent jursiprudence.
If you think rights are everything outside of the necessary intrusions of government, yes.