At my club, we ask for name, address, license number, e-mail, and phone numbers. Some ask for a description of your car and tag number. That's so that if a member sees someone doing something wrong, but can't get the person's name, they might be able to track them down by vehicle.
We also ask for your line of work (but not your employer). We want to know if you are plumber, so that if we need to fix the bathroom we know who to call who'll give us a break on the cost.
We ask if you have ever been arrested. We did have someone try to join who had an FID, but was a federally prohibited person. As you might imagine, that was a rather uncomfortable interview when we turned him down, but we did not want to be involved in that, even tangentially.
We ask a few other questions about your interests. We're trying to get to know you quickly, so that we can vote on your membership.
As for SSN, no we don't ask for that I certainly wouldn't give it out. I suspect its still on there simply because someone thought it was a good idea way back when. Join the club, go to the club meetings, suggest to them that they should not be collecting SSN, and offer to draft up a revised application form without it.
In terms of who has access to the information, our board members get a paper copy of the mailing list, with names, addresses, and phone numbers. The files are kept in a locked office. The mailing list is kept in electronic format, but behind hardware and software firewalls on a machine that is not always on. We do not release our mailing list to anyone -- I suppose we would if someone came with a subpoena, but that's never happened.