I don't think as things currently stand that you can disable it, but its name is apparently "Context Switcher".

According to this MSDN blog, it would appear to be a new feature intended to help you manage shared files in Universal Apps. I agree that it's confusing, and I'm not a big fan.

A thorough search of the VS options for anything related to "Navigation Bar", "Context Switcher", or "Universal Apps" comes up empty and there don't seem to be any extensions offering this capability either. As far as I know these would be the main avenues for configuration, so my conclusion is that we are stuck with it until the next VS update or until someone gets around to making an extension that can disable it.

If you prefer, you can disable the navigation bar entirely in "Tools > Options > Text Editor > All Languages > Navigation Bar" (or you can disable the bar on a language by language basis.)

Update: As of Visual Studio 2013 Update 3, you can drag and adjust the relative sizing of the 3 drop down lists in the navigation bar.

From a little local testing it looks like the sizing you set is shared between all files and solutions and it persists after closing and reopening visual studio.

I shrank the context switcher down to just the visible text, and it feels more well proportioned and closer to the classic Class and Member drop down layout.