If the piece movement rules are randomized, it's not going to be learning chess, is it...

Maybe they meant the piece type is randomized, because each piece has a different movement rule. This would result in things like maybe 5 pawns and 3 rooks instead of the normal numbers.

There is some discussion on reddit:

https://www.reddit.com/r/chess/comments/c4dgas/alpha_zero_fi...

That would be Fischer Random Chess (aka Chess960). I know engines play it, but I don’t know if they’re trained on it.

That indeed randomizes the starting position (subject to some constraints), but it doesn’t randomize the abilities of the pieces.

AlphaZero has been used to more objectively study chess variants in this paper: https://arxiv.org/pdf/2009.04374.pdf

I don't think they mention anything about speeding up training. The original training time for AlphaZero was only a few hours anyway so I don't think that was ever a major constraint. I would imagine that each neural network performed best on the variant on which it was trained.