Unlike the Roku, the Shield TV does not support Miracast so there's no native way for Windows to cast to it.
Your best option if you want to extend the desktop is to get a Miracast adapter for the TV, or even use your old Roku as a receiver. If you don't mind mirroring, and don't want to install Chrome, you can use Edge Chromium (turning on "Load Media Router Component Extension in edge://flags") to mirror the desktop to any Chromecast compatible device.
Unlike the Roku, the Shield TV does not support Miracast so there's no native way for Windows to cast to it.
Your best option if you want to extend the desktop is to get a Miracast adapter for the TV, or even use your old Roku as a receiver. If you don't mind mirroring, and don't want to install Chrome, you can use Edge Chromium (turning on "Load Media Router Component Extension in edge://flags") to mirror the desktop to any Chromecast compatible device.