What numbers, all I see is the years

understandable.

I do not feel that connected to any empire, but if you talk in today terms, what is india, was not really a 1 big soverign entity. It was much like todays europe - tonnes of states. One could argue mughals were the most recent empire under which a very large part of current India was under (almost entirety of north and west and central). they had a larger cultural impact. Our current most spoken language (hindi) did not exist, lets just say, 200 years ago. there were tonnes of derivaties from older sanskrit and dravidian roots (european equivalent would be romanic and germanic roots). what mughals brought (not them, it existed before as brought by trade and previous empire (tipu sultante, which is very small, but still some what important)) were the languages usually used in islamic world - arabic and persian (farsi as we call it). from these older languages and their amalgamations we got "hindustani" . Consider it parent of hindi and urdu (most spoken languages in northern india, and pakistan). these 2 languages are in some sense same - 2 speakers can communicate orally, but not in writing, because urdu reatined the arabic script, and hindi retained the devanagri script (sanskrit). Mughals had a large enough impact on india, something our current administration really tries to downplay, and paint in very bad light. Yes, they were not the ideal, utopic, do-no-evil kingdom. But no empire has been that, ever. they were very comparable in terms of other empires in terms of social standards.

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

It drives me crazy that we haven't translated their written language, and know none of their stories, none of their heros, philosophers, or rulers. What we have gleaned from their artifacts suggests a marvelous society rich with tales, and every single one of them is currently lost. I need to go visit their cities one day.