In case you’re just getting started with Auth0 and if your application is a SPA (Single Page Application), I suggest to look at the Auth0 SPA SDK instead of the (older) Auth0.js (which you linked above).
In regards to the connection. You can create any other database connection (i.e. in the Auth0 Dashboard) and use that instead. You can give it any name you like. Renaming a connection isn’t possible though.