If you want a highly dependable solution to wirelessly display your Microsoft Surface 3 but also works with other devices, the Belkin Miracast Video Adapter is a good choice.
In this example, I hooked up the Belkin Miracast Video Adapter to a Philips TV in a hotel I was in. Identifying the Belkin Adapter and connecting to it was pretty quick.
One of the nice things with the Surface 3 compared to other tablets is that similar to what you are similar with seeing on a PC, there are a lot of options on how to use the mirrored screen. Using one screen or the other, duplicating the screens, and extending the screen all worked well between the Surface 3 and the Belkin Miracast Video Adapter.