There may be other questions. They may want to know your date of birth. It's none of their business. They'll say they need it to make a positive identification. Again, Scripture says, "Do not cast your pearls before the swine." When Jesus was being accused by the chief priests and elders, "He answered nothing." Matthew 27:12 You don't have to say anything. You do not have to help them in any way. Remember, these are the very people who are trying to put you in prison for a long term, why should you help them? They want to destroy you. Do you want to assist them in their destructive force? Only someone who is mentally ill would want to help them in their destroy mission.
As far as a social security number, I don't have one. I assume you don't have one either. You can't give them a social security number if you don't have one. One time I heard a police officer come to the rescue when a social security number was being demanded. "Leave the guy alone," said the officer, "maybe he doesn't believe in social security," and that ended the controversy. You have to remember that there might be a few people in the system who sympathize.
One of our citizens had a standard answer to their questions, "The Church has not authorized me to release that information." He recognized the Church had sent him and that he was under orders. It frustrated the jail personnel because he would not submit to their authority over him. After 5 days, he was released without submitting to booking procedures or signing any agreements.