It stems from the president's constitutionally granted pardon powers. SCOTUS decided this also includes the ability to grant amnesty.

It's a petition for amnesty because he hasn't been convicted of a crime, yet. Colloquially the two words are pretty much interchangeable, but pardons and amnesties are different things.

It would probably be better described as amnesty. I know the word pardon was used in reference to this, probably because it derives from the president's constitutional pardon power, which was interpreted by the SCOTUS to also include granting amnesty. But the petition should really be a petition for amnesty, not a pardon unless he's successfully convicted and all avenues of appeals have failed.