No, you don't always have the right to say nothing at all. Courts can compel testimony and punish you if you don't.

And you plead the 5th after going under oath. And you can't just plead the 5th to any question. If the prosection puts you under oath and asks you your name, you can't plead the 5th to that

That's why I said generally - once testimony is compelled, it can no longer be used against you. And the definite exception for compelling your name is if the government already believes that you committed a crime and is trying to figure out who you are, and you cannot articulate specifically why your name could be incriminating.

5th amendment protections can include questions of identity, if the question of identity is relevant for incrimination. Like, if the government has a warrant for "Joe Smith", you're not required to testify whether that's you. It's usually a waste of time since could just prove it with the non-testimonial evidence that lead to your arrest, but the protection does exist.