The purpose of our VsDialogWindow.cs appears to be that, in the past, the VS platform assembly name was revved for each product iteration, meaning our XAML code needed a wrapper to target the correct assembly without specifying the version# in XAML.
The version is no longer being revved for Microsoft.VisualStudio.Shell.15.0.dll, so I believe this can be removed entirely. We also have other direct XAML references to this assembly already (eg, InfiniteScrollList.xaml).

I found a comment here that makes me more confident that this is safe to remove:

/// <summary>
/// Works around an issue w/ DialogWindow and targetting multiple versions of VS.
/// 
/// Because the Microsoft.VisualStudio.Shell.version.0 assembly changes names
/// we cannot refer to both v10 and v11 versions from within the same XAML file.
/// Instead we use this subclass defined in our assembly.
/// </summary>

https://github.com/microsoft/PTVS/blob/bcdfec4f211488e373fa253583fefe3dbb320c87/Common/Product/SharedProject/DialogWindowVersioningWorkaround.cs#L20-L26