MacOS: Add browser tab (i.e. Gmail) to application switcher?
Use Fluid:


  Fluid lets you create a Real Mac App (or "Fluid App") out of any website or web application, effectively turning your favorite web apps into OS X desktop apps...your chosen website has a permanent home on your Mac as a real Mac application that appears in your Dock.


Gmail is even one of the examples they show in the video, and it supports features like badging the application icon when you have new messages.