If you use a DEX on a blockchain like Ethereum or Binance Smart Chain (BSC), the DEX can only interact with tokens that are created on that particular blockchain. Ethereum does not "talk" to other blockchains such as Bitcoin and BSC, and vice-versa. In other words, an Ethereum transaction cannot modify the state of the bitcoin blockchain, they are both independent of each other.
So how come bitcoin, doge coin and all those foreign tokens can be traded on Ethereum DEXes?
The answer is that they are wrapped tokens.
In short, someone put X amount of bitcoins in a reserve account on the bitcoin blockchain and issued an ERC-20 token on Ethereum with X tokens in circulation (such as WBTC). The ERC-20 tokens can be exchange 1 to 1 for the bitcoins in the reserve account. The DEX will trade the ERC-20 token only, not the tokens on the native blockchain reserve. Here is a more detailed explanation by Binance.
Wrapped tokens are generally somewhat centralized, meaning that some entity somewhere has the private key to the native blockchain reserve (the bitcoin account in my example). Consequently, they carry an additional risk compared to holding the native token directly.
An exception to this is when a token is wrapped on the same blockchain where it is issued. An example would be ETH - WETH, and BNB - WBNB. These wrapped tokens are wrapped by a smart contract not controlled by anyone.