It's "real", but somewhat deceptive. The melodic line (the notes that he's playing one-at-a-time with his right hand) does indeed map to the decimal digits of $\pi$. However, the harmony (the notes that he's playing with his left hand), are just arpeggios in A minor and have nothing to do with $\pi$. If you removed the harmony and just listened to the melody, then you probably wouldn't find it to be particularly aesthetically pleasing, and probably no more or less so than a completely random melody composed from the same ten notes.