If you don't have one yet, the easiest ways to get started are to use the Rainbow mobile app or the Metamask desktop browser extension.
After you have a wallet, you can connect to this site and start chatting. You will be identified only by the public wallet address you connect with.
You can use the same Ethereum account on multiple browsers or on a laptop and phone. You could try creating multiple wallets and switching between them. Creating multiple wallets can be done in any of these applications though it is easiest on desktop with Metamask.
This site is just a demo of Sign-in with Ethereum. The messages are not sent to the blockchain. The wallet and your Ethereum account are only used for authentication. No personal information is collected or stored in any way.