Hello!

What is the Difference between Chat module and Messenger module?  Is one more mobile oriented?

You may read about differences here https://una.io/page/view-post?id=32 in more details.

In our installation, we have messenger and video call, but when site is loaded on android I only see text messenger.

In UNA mobile app you should see Video Call button. If you are using standard browsers on mobile devices Video Calls will not work, because jitsi doesn't support this ability in mobile browsers and works on through the mobile apps. 

Is there a separate android app for messenger? Or should it be used on a mobile device as a web page?
If there is an app, how do we create a branded one to work in concert with our UNA website?

UNA has own mobile app which allows to use Video Calls on mobile devices.