I'm pretty sure there is no anonymous mode allowed for Mode S units.
A quick search turned up this link: https://www.faa.gov/nextgen/equipadsb/resources/faq/ from which I pull this quote (I added the boldface to the last few sentences): "ADS-B Out equipment transmits information about the aircraft's location, ground speed and other data once per second. The broadcast includes the aircraft's unique ICAO address, which for a US registered aircraft is associated with its tail number, as well as the transmission of aircraft's flight ID, which for GA aircraft is generally the registration "N" number, or for commercial/government operator's, their call sign or airline flight number. Air traffic controllers can immediately see this information if they desire, however the controller has display options where they can suppress the N-number for VFR aircraft that are not receiving flight following services, and some controllers will set these options to "declutter" their display. Additionally, GA operators that employ UAT transceivers have the option in VFR operations, when they are squawking 1200 and flying below FL180, to select "anonymous mode", which enables the transceiver to broadcast a randomized flight ID and pseudo-ICAO address. The transceiver reverts back automatically to the assigned ICAO address and default flight ID when the beacon code is changed from 1200."
But as far as the reason goes for implementing it at all, I believe it's just that not everyone is comfortable with the government tracking their every move, or getting a letter from the FAA due to some accidental violation, such as a brief incursion into Class B for instance. I get that. Oh, that brings up another thing: I'm pretty sure I read somewhere that you can't use ATC services if you're in Anon Mode. Makes sense. [Edit: duh, you're squawking 1200] Also, I'd worry about if you crashed. It seems like it would be easier to find you if you were not anonymous -- they could just look up your flight track by your tail number and see where it ended. [Edit: FAA specifically mentions this as one reason to have ADS-B Out.]