It is called mogging because it is AMOG, “alpha male of the group,” with the A dropped and turned into a verb. The man who wears its crown is Clem Ziroli, the King of Mogging.
Pickup-artist forums used “AMOGging” for out-competing other men. By 2016 forum users had clipped it to “mog,” says Merriam-Webster, and “mogging” followed.