In the US and when sending packages within the US:

You have to put a return address on, and they may validate it at some point to the extent that is has to be a real address, but you do not need to show ID or prove that it's yours. But obviously, you'd need a return address in the same area as you're mailing it from, so it, along with postmarks, would still reveal the general origin.

You do not need ID for mailing packages, though you will get asked to verbally verify that there's nothing liquid, fragile, perishable, etc (also no batteries).

You can also use services like Pirateship to buy and print postage and then drop it in the package drop-offs at the post office. Pirateship will require enough info for you to use a credit card, but as with the post office, the return address just has to be one that exists, not one you have to prove is yours.