You really want to ask a lawyer who is versed in tenant issues this question. Probably good to have a copy of whatever legal agreements you signed when you joined the coop, and to have read them. And it wouldn’t hurt to have a copy of your insurance when you go. It looks like you are in NYC, so you should have low cost legal services available if you do not have a go to person you are already using.
It would also be helpful to send a written request to the management asking them for their reasons for asking for this information.
Good relations are really helpful in any such relationship, so if you feel strongly that providing that for which they ask would hurt your relationship of confidentiality, or there is some other reason for not providing it, having absolute legal advice that you do not have to do so first will be helpful. It would really suck to say no, and then later have to give it to them anyway.