If my bathroom got flooded by neighbor upstairs, is it legal for them to withhold their home insurance information from me?

Yes, it is legal, until and unless you sue them. Their insurer is not there to protect you--their insurer's duty, and the reason the neighbors pay their insurance premiums, is to protect them. Neither the insurer nor the neighbors have any obligations to you, and the neighbors do not need to make it easy for you to get compensation. 
That does not mean that you have no rights. Your recourse is to sue the neighbors, if you feel that they were "at fault" in causing the flooding--e.g. that they were negligent or careless in some way, such as by turning on a bathtub to fill then walking away, or flushing unflushable things down a toilet, causing it to back up. If the neighbors were at fault, they may be liable, or financially responsible to pay for, your damage e.g. the repair costs. If you sue them and prove their fault, you can get a court judgment requiring them to pay. If you do, either they or their insurer depending on their policy coverage will have to pay you. During the course of the lawsuit, you can get information about their insurance and they or their insurer may well elect to settle once you sue, and they see that you will take this all the way to court if necessary.
Remember, though the neighbor is only liable if at fault. If they did nothing wrong in causing the leak--e.g. a pipe leaked or burst through no fault of theirs--they are not liable. If they are not liable, their insurer also will not have to pay, since their insurer does not insure you, but them, and only pays if they have to pay. This is  why you should have your own insurance to pay for damage or repairs, because sometimes there is no one else liable to pay for you.