Back in the 70's, I had a friend who was a big fan. One thing we noticed while looking at the pictures inside the album was that there was a name under every picture except the singer. To our drug-addled minds, this seemed to fit with what we'd heard, that "Alice Cooper" was the name of the band, not the person. We decided that the band was those four guys, and the singer didn't have a name, which we thought was really fucking cool, even if we were way off base.
It also seemed to fit with the line "She asked me why the singer's name was Alice. I said 'Listen baby, you really wouldn't understand.'" It was more complicated than that.
But naturally the idea that "Alice Cooper" was the band and not the person didn't work for most people. Too meta, too weird. So the previously unnamed singer (nee Vincent Furnier) became Alice Cooper, and when the band broke up, he went solo and called himself Alice Cooper. That's how we always understood it, anyway.