So does every cryptocurrency including bitcoin and altcoins have their own different type of blockchain?
No, Not every cryptocurrency has it own blockchain.
The one that has it's own Blockchain generally called Coin, the one that do not have their own blockchain called Token.
The one that does not have their own blockchain use the blockchain of crypto that run smart-contract, some of the are Ethereum, Binance Smart Chain, Stellar. etc.
Some coin that initially does not have it's own blockchain, later on their development end up creating their own blockchain and then migrating their project to their own blockchain.