Then why call it the Eastern Roman Empire, or the Byzantine Empire if it's the same thing?

Because it's helpful to compartmentalise something with so much history. We can refer to Capetian France and Valois France even though they're both still France, for example. It is unusual for such compartmentalisation to be used with a geographic aspect rather than (only) temporally, but the Roman Empire was unusually big, especially for its time

The two were, of course, independent from one another for basically all practical purposes. I just also think that both are equally the meaningful continuation of the united empire

In very technical terms, the split was a legal loophole to maintain a "single" empire on paper for their own egos and to avoid conflict. In practice and in hindsight, they're 2 different things.

Surely if we are going with the principle that only one of the two halves of the empire is the real continuation, it ought to be the one that kept the capital and administrative structure of the unified empire? Or if neither is the real continuation, did the empire die under Diocletian and then Constantine ruled over a separate entity altogether after the end of the tetrarchy?

Does the British Empire still exist? I would say no.

To keep the analogy accurate here, we'd have to make a few changes. The British monarchy does not call itself the "Southern UK" - you pointed out yourself that the Romans did not call the ERE a different thing - and it'd also have to keep control of a substantial chunk of the colonies when Britain falls to Germany. I do think that, in that case of George V moving to Australia and continuing to rule, say, New Zealand, India, and South Africa from there, I would still consider it the same empire. This exact situation actually pretty much happened in real life with Portugal and Brazil, it just didn't happen in advance of the fall of Portugal and didn't last very long.

Other than this would make some great alternate historical fiction.

I think one of the big HOI4 mods, Kaiserreich, actually does do this! Although as I understand it, it was due to a communist revolution after a stalemated WWI. I do not know the details as I've never played it, I only found out because I had been chatting about ideas for an alt history WWI outcome with a friend and we found out that we had basically accidentally come up with Kaiserreich's version of Europe