Is there a way to show two Metro apps (the kind you launch by clicking on a tile) at a time in Windows 8 as opposed to filling up the entire display every time you launch one?
Not in the same way as regular Windows applications.
This is only a partial alternative, but works well in certain scenarios, particularly when you want to have browser windows side-by-side.
There's a new app called Toolbox for Windows 8 that offers many different "snapped" views within its own app. It has its own nine "tools" built-in, including a browser:
These tools can be arranged in a variety of pre-defined snapped layouts, and can be put together into custom layouts as well. One of the options is a split side-by-side view, which is useful for viewing two webpages at once, for example:
You can also put together multiple tools into custom configurations:
And you can even split tools while snapping the whole app to one side, letting you snap within your snaps. Yo, dawg!
Obviously this won't work for all situations, but I've found it to be a very useful tool (no pun intended) for situations where the default options of full-screen or 33%/66% don't quite cut it.