That's a bit of a broad generalization. As with all history it doesn't hold up under scrutiny.
In the 1200s, the empire split into 3 warring countries. The one that took Constantinople was considered the continuation of the Byzantine empire, but maybe it's more correct to think of it as a spiritual successor, than an actual unbroken reign of a single nation.
Which would put it at less than 1000 years of rule. So "a few hundred years" would still be an accurate upper bound.
I really don’t know much about it, but I think the Han identity is similar to whiteness in the US, right? A coalition of groups that not that long ago would not have considered themselves the “same.” The Chinese govt seems to consider constructing a unified Chinese identity pretty important, maybe anticipating that once they start failing to make the standard of living gains they’ve made over the past decades ethnic tensions could be destabilizing?