You don't have to provide anything to make a wallet, but you do have to give banking details to whomever you buy the tokens from, and as of the boom a few years ago, any exchanges that don't want to be cut off from the global banking system will be as thorough about verifying and retaining data about your identity as any other regulated bank. Proper obfuscation would require something like buying Bitcoin for cash on the street and doing some prosecutable money laundering for good measure.