There are hundreds, if not thousands, of blockchains. The number will continue to grow as blockchain technology is adopted and adapted to new use cases. In time, there could be hundreds of thousands or millions of chains in the same way that there are so many computer networks in the world today.
There are different types of blockchains. There are public and private blockchains. There are blockchains that have only one coin or token in use. There are blockchains with lots of tokens on them. And there are chains with no coins or tokens.
It can sometimes be confusing as to which is which.
Public projects with their own 'native' coins usually have their own blockchain. Examples are Bitcoin, Ethereum, Litecoin, Dash, Zcash, Monero, Dogecoin, Decred, Horizen, DigiByte, Ravencoin, etc. These are all separate blockchains.
Many projects don't run on their own blockchain, but on another upon which they are built, usually on smart-contract platforms. ERC-20 tokens, for example, run on Ethereum. This includes BAT, Decentraland, ChainLink, DAI, USDC, Enjin, Aragon, etc. They all run on the public Ethereum blockchain.
The advantage of being a token on another chain is that you get all the security and stability benefits of the parent chain and ease of setup by following a standard.
In addition to Ethereum, other smart contract platforms include Ethereum Classic, Tezos, EOS, and Cardano. New tokens will emerge on these blockchains as projects and use cases are born that leverage the unique advantages of each chain.
There are also private chains that don't have a coin or token, but track information for particular business purposes. These are often built on Hyperledger or Corda, where the chain is only accessible to 'permissioned' users. We may never know how many of these exist as they are akin to (or used on) private networks that aren't exposed to the public.
Public blockchains have coins/tokens and mining/minting to incentivize participation in support of the network.
Private blockchains typically don't have coins or tokens because they're financed by corporations and their partners who run those chains in support of their core businesses.
In addition, the high number of blockchains each with its own varying features, protocols, and underlying infrastructures can adversely reaching the critical mass of users. As it gets complex, user interface can get perilous.