I imagine that the biggest problem with maintaining an identity would be if you were collecting social security long after you should be dead. Even if you were one of those people who always looked older than you were, as a wizard you would look way younger than 100+ when you were 100+. If the government decided to investigate your collection of social security, they'd be sure you weren't you. Otherwise, as long as you're paying taxes, the IRS is unlikely to investigate.
What do you need an identity for? Paying income taxes. Collecting government benefits. Travel. Buying booze. Buying guns. Getting credit? Any sort of licensing. Banking.
Plenty of that could be solved by getting id from, and banking in, a foreign country. Travel is already a problem, and the solution is to "cheat," as Dresden puts it, using the Ways. Then there's just breaking the law by not paying taxes, black market gun buying, and just not obtaining the "necessary" licensing. Trusts, llc's, lp's, corporations, and such would be very useful for a lot of these problems, but you would still run into tax/tax id problems.
All of this is probably moot because I think the White Council has paperwork for maintaining an id with the government. I base this on Rashid talking to Dresden about fixing his presumed dead situation with the government. I know that's a little different, but I assume the Council has procedures for this because the Council wants to maintain secrecy, and it has to be an issue that has been around for a while. I imagine it's the type of thing they want to stay on top of.
On the other hand, a lot this has only come up recently. Money laundering only became illegal in 1986, which is a much narrower thing than I previously thought, but there are all sorts of regulations that have come up after that to hinder money laundering and tax evasion that would get in the way of maintaining your official financial existence. Then a lot of new laws are coming into effect now to combat fake ids because of 9/11.