As far as I know (not tested), the files are simply stored on the external storage connected to the internet box.
The router doesn't have to support it. But your TV has to support the connection to a file share (or NAS).
Perhaps your TV accepts only local storage like USB but doesn't like remote shares coming from a NAS.
If you have a swisscom TV UHD Box, than you can play a lot of video formats with the built in VLC app