mannequin.setPlayerProfile(player.getPlayerProfile()); does not texture the mannequin the texture of the player. The resulting texture of the mannequin is the default skin "Kai".
 
The full code that I ran was the following:
player.setGameMode(GameMode.SPECTATOR);
Mannequin mannequin = player.getWorld().spawn(player.getLocation(), Mannequin.class);
mannequin.setHideDescription(false);
mannequin.setPlayerProfile(player.getPlayerProfile());
mannequin.setImmovable(true);
player.setSpectatorTarget(mannequin);
This resulted in the player spectating a mannequin that did not have the same skin as it. The player is online when this code is ran.