Yes it is possible. The easiest way is to download the Yalp Store app ( https://github.com/yeriomin/YalpStore ). In the settings you can go to "Pretend to be a different device" and choose the Nvidia Shield. This will allow you to download the TV versions of the app.
Depending on your launcher some apps might open as the mobile version. You can circumvent this by using HALauncher or there is a modified version of the Lean back Launcher available here https://github.com/rockon999/LeanbackLauncher/releases that works pretty well. Good luck!
I'm pretty sure that Android TV Google Play Store apps are actually a subset of the main Google Play Store
Any specific apps you are considering?
There are only a few apps that are uniquely designed for official AndroidTV
I tried installing Live Channels apk on my phone....wouldn't work
Make sure prefer TV mode is checked in the settings. If it's still not working I'd give the leanback launcher from XDA a shot. If it's still not working it's possible the mobile version was downloaded. https://imgur.com/j0ZQ9FF
Few Android tv apps can be installed on any android if you have this app cloner and select Android tv in options https://play.google.com/store/apps/details?id=com.applisto.appcloner