When you enable both the database and Azure AD connection the Lock screen is still showing an UI that allows you to login with both connections; it’s just not the same UI used when Azure AD is the only available connection.
For enterprise connections, Lock has built-in support for Home Realm Discovery (HRD) based on the domain part of user entered emails. If you enable both the database and Azure AD connection and then proceed to enter an email from a domain associated with the Azure AD connection then the Lock updates itself to reflect that. For more information, see:
Also note that this behavior, as far as I’m aware, is not configurable so if you enable the database connection plus the Azure AD one you will get the streamlined UI and no button specific to Azure AD; you’ll only see the button if the Azure AD is the single connection.
If you need more control over the UI aspects the recommended approach is for you to use Auth0.js, which is a UI-less library, instead of Lock.