Moving our own ERC20 tokens crosschain

Hi everyone, im a former student of IvanOnTech, and wanted to ask some tech help.

Lets say I made an ERC20 token on BSC called “MOON” , minted 1million of them and they are successfully trading. Now I want Polygon to also have my MOON tokens, and for BSC folks to transfer their BSC MOON to Polygon.
Do I have to deploy an ERC20 contract on the Polygon chain? If yes should that contract also start with a balance of 1million?
Or is there a different road to take?

Any advice greatly appreciated