It depends on the music. Game grumps generally use music from old video games. It's music that belongs to game itself.
Let's imagine this scenario:
The new FIFA 2014 will include a song from Skrillex. FIFA Developers talked to Skrillex and negotiated with him in order to be able to use the music on the FIFA game.
Even though the music is being played on your console, the music still belongs to Skrillex, NOT the game.
If your doing a let's play of a game of FIFA and the Skrillex music from the FIFA game pops up, you're making a copyright infrigment.
Why?
Because the owner of the content decides if their content is free to use or not. FIFA Developers allow all uses of the game content that they own TO ANYONE (as long as it's displayed along with the game). Skrillex does NOT allow use of the music that he owns.