Since several people saw the body and identified it as that of "Billy the Kid" (Henry McCarty, William Bonney) it's quite likely that Billy the Kid really is buried in Billy the Kid's grave.

HOWEVER -- there were several people in Ye Olde Weste who were nicknamed "Billy the Kid" both before and after the one Pat Garrett shot. Pat Garrett was shot while taking a piss along the side of a road, probably by Jim "Deacon Jim" Miller -- another of the psycho killers.