Client Logo in Hosted Page lock widget?

It would be technically possible, but not aligned with what I mentioned before about the notion above that the hosted login page should not be branded by client application, but by the service to which it is associated.

For example, if you service example.com you would have an Auth0 tenant like example.auth0.com and ideally you would have a custom domain identity.example.com that points to example.auth0.com so that the URL end-users see when accessing the hosted login page is in your own TLD.

With this in mind, the branding of the login page should be all about the service Example and not about a particular client application that is defined within that service. It would be normal to have different client applications to handle web and native scenarios, but the brand should still reflect the Example service.