The shell groups windows in the taskbar using each window's Application User Model ID (AppUserModelID).
By default, every window generated by a given EXE (even in different processes) shares a system-generated AppUserModelID.
You can give each process its own AppUserModelID by calling SetCurrentProcessExplicitAppUserModelID. From your description this is probably what Skype is doing, though I haven't checked.
You can give each window its own AppUserModelID by setting a different PKEY_AppUserModel_ID property on the windows.
Note that these IDs are required to have a particular format:
CompanyName.ProductName.SubProduct.VersionInformation
Raymond Chen wrote an article about this, and it's also worth reading the documentation I linked to.
I'm not aware of WinForms having explicit support for this, but you could certainly use interop to call the Win32 API directly.