No reason to get into the science of canine perception of images on small-scale video screens, just as there was no reason to fret over the “universal translator” in StarTrek. They are necessary plot devices to help move the story along.
It is also surprisingly silly to assume that Rex could not have brought his dog with him while doing his show. I’m sure that kind of thing happens all the time. In fact, my minimal research confirms it is quite common. And how many times have we seen Mark “chat” with Andy on his smartphone? Irrelevant, as this is just another plot device, put here to provoke male bonding.