I'm sure they get fake identities the old fashioned way, the same way vanilla mortals do now. They have to provide proof of identity to go to Univeristy in any country, and most of them have a lot of money to buy those kinds of things.

If regular humans are able to create fake identities for a lot of money, I'm sure wizards can too paying the same people. And they mention at one point that wizards don't have any limits on using magic to make money as long as they don't break the laws of magic. So they could do some outright unethical things in our eyes if they keep to the laws. Harry and Ebs talk about it at one point in Changes I think.