If you want the same profile dropdown menu displayed on the desktop view, please be advised that it is not available on mobile view. 

This is the intended behaviour that our developers designed for the mobile responsive view. The breaking point of the 799px width display will hide the profile dropdown menu, which will make our BuddyBoss Theme mobile responsive and adapt to different displays our developers designed

You may need to do custom work with the help of a personal developer as this will change the functional aspect of our product.

This is a great idea, and we highly suggest that you submit this to our, Roadmap as a feature request. While not guaranteed, if it gets added to the product, it will be listed in the release notes of the latest update and other resources.