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Hey there, let me answer your questions one by one.
>If I transfer USDC (kept on Binance) to ledger Nano S Plus via Polygon Network for cheaper fee, is it safe or should I just use Ethereum Network to do the transfer?
You can consider it safe yes, only make sure to always have your 24-word recovery phrase secured, all your account addresses are derived from it.
>Are the USDC tokens sent via Polygon Network remain the same USDC after transfer or they change in some way? And which blockchain will hold it after such transfer?
They will hold the same value only the contract address will be different. If you transfer the USDC using the Polygon network they will live on the Polygon blockchain.