Earlier quoted context omitted.
Plenty of games have had elimination modes for a long time. This is really just an evolution of that adding the mechanic of the map edges closing in.
Elimination mode is more generic. The battle royale subgenre started with the Minecraft hunger game mode, and grew more popular by H1Z1: King of the Hill, and then by PUBG.
Minecraft had nothing to do with it. PU was inspired by the Battle Royale novel/movie and The Running Man. He forked the Day Z ARMA2 mod and made DayZ: Battle Royale, and that predates H1Z1 KotH by 3 years.