Sending USDC from metamask to exchanges

Hello I got some USDC in Eth network I metamask.

Every time I try to move it I need to pay gas fee ( sending usdc to binance or kraken)

If I send my usdc to binance (example) under Eth network. Can I send usdc back to metamask under matic chain? (Instead of bridging usdc Eth to usdc matic in metamask)

Or even if it is in binance my usdc it still in Eth network?

Thank you all and apologies if this has been asked before.

I have been in the space for a while but normally I just deal directly with exchanges.