Suppose that you open an account at an exchange and buy 10 bitcoins. Then the exchange assigns to you an address but keeps the private key. Let say C1, C2, C3, ..., C10 are ten different coins (tether, etc.). In the next 10 hours, you trade these 10 bitcoins for every C_i coins, one per hour. Should we expect by the end of the day we will see my address (the one assigned to me when I opened the account at the exchange) shows up 10 times on the bitcoin blockchain? (In fact, for every transaction, there is one unit of bitcoin flowing out of my account and one C_i flowing in. There must be changes in the blockchain of both coins. I am asking about the bitcoin blockchain. )

Or the exchange handles these trades internally and nothing happens on the blockchain? If latter is the case, then once in a while I should see the exchange makes a huge transaction on the blockchain, right? Is this why we see those many "multi-sender multi-receiver" transactions on the ledger?

I can go and get a bitcoin through blockchains (on-chain). Also I can go to any exchange (say, Binance) and buy one (off-chain). All I know is that they are different at least with respect to confidentiality (to some extent; if one knows my wallet address on blockchain, they can follow my transactions over time but not on exchange). My question is more about the difference in nature of the two coins. I imagine the legal rights can be different.
What I get on Binance is probably just an IOU instead of an actual bitcoin. For example, is there a way to transfer that bitcoin from Binance to a ledger, without cashing out the position in Binance and buying one on-chain?
In other words, are on-chain and off-chain bitcoins (or other cryptos) identical (or fungible) animals or is there a contractual arrangement by the exchanges that make them identical (for example some type of pegging mechanism inherent in the exchange design). any reference to related articles also helps