are exchanges have another type of blockchain running?
What do you mean by running another type of blockchain? We have only one bitcoin blockchain.
I think OP has messed up Coinbase exchange and coinbase transaction.
To OP, if my above assumption is correct, let me share with you that these two are different. There's an exchange namely "Coinbase" but the block reward generated by the miners in each block is also called coinbase transaction. Hope this explanation clears up your confusion. Many people get confused with these two terms lol,. you are not the only one.