I have no id. I think it is just the style
Why the Bishop has cut on it's head?

bishops look like frowning depressed pieces 💔
lol. Which pieces look happy to you?
none, the game is meant to be what it is which is a war where you kill someone and make them bleed until they are finally gone or choose to bleed out from the pain

The bishop piece looks absolutely nothing like an elephant, though. I can't even see a tusk no matter how hard I try.

Or could it have been this exposé
https://www.chess.com/forum/view/off-topic/the-sad-tale-of-the-pawn
Rooks are not siege towers. They are chariots. The original name for chariot was similar to the Italian word for tower, so they became towers. The bishops are elephants and the rooks are chariots. Thus if you make those adjustments, you have a real army. The royalty protected by war elephants, knights, chariots, and soldiers.
The Bishop has a cut on its head because of the Protestant Reformation.
I didn't know this. Thank you very interesting.

Where is the Chariot in Indian chess?
It's drawn by Camels.
... or maybe the Elephant stepped on it.
Yeah, and don't say the Rook.