It's not really an additional installation you're encountering. When games for the PS4 or One install, most will allow a player to play while the installation is ongoing, typically the first level or two, so the player can get started on the game quicker and with the assumption that by the time they finish the first area, the rest of the game will installed.
For those that don't allow play before the game is installed, like GTA or Overwatch, it's because they really can't break off a chunk of the game for early play, so the player has to wait until the installation has finished.