Is the Pi melody real or fake?

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.

Here is an alternative take on mapping $\pi$ to the diatonic scale. There are probably as many ways to translate $\pi$ to melodies as there are digits in it. And all mappings from digits to notes are arbitrary, as is the base 10 representation of numbers. It might be interesting to use a base 12 representation and map to the chromatic scale.

There have been attempts at using $\pi$ in the construction of tuning systems. The so-called Lucy tuning has a fifth of $600 + 300/\pi \approx 695.5$ cents (in tempered tuning it is 700 cents). A search for this tuning system may soon take you into New Age land, so be warned.

This is what can be made from the digits of pi base 12 played in the 12 tones.

https://www.youtube.com/watch?v=0Dwvn73tz8E