I don't know about the United States as I've not been, but in every other country I've gotten a SIM it's been a matter of picking it up and the store and handing over money, same as if you're buying, say, deodrant. You can top up money anonymously too by buying prepaid vouchers.

I've certainly never had to hand over stuff like social security number (wtf?), but perhaps laws and/or telcos are different in U.S.