I am writing some Powershell scripts that are going to be in a Nuget package, which I will then run from the Package Manager Console. These scripts will run all the tests within the project that is selected in the "Default Project" drop down, in the Package Manager Console. My question is, how do I determine (in Powershell) which project is currently selected in the Package Manager Console?

There are a number of useful commands available to you in the Package Manger Console, one of which is:
Get-Project which on its own will give you the name of the Default project in the Package Manager Console. However:
Get-Project -All Produces a list of all the projects in the currently loaded solution.