This is such a thorny subject I feel, for clarity but also for my own safety, the need to start by pointing out that I am myself what we in the UK call mixed-race. But I spent the first half of my life in North America, various bits, before coming to the UK.
So, let's start with some basic facts:
When Barack Obama first became a presidential candidate, our UK newspapers all referred to him as "mixed-race." Only when he made it clear he identified as black, or Black, did the UK papers switch. It took about two weeks.
With Meghan Markle, there was also a bit of back-and-forth, but it was much shorter, possibly because she made her own view clear quickly. What weirded me out a bit at first is that not only does she straighten her hair but she clearly (look at pictures of her over time) uses skin-lightening cosmetics and may even have had a nose job -- and why shouldn't she? In Hollywood I bet that was a survival tactic.
Kamala Harris's father is Jamaican and always as far as I know identified as black, and she attended a historically black university, Howard. However, looking at photographs, he himself clearly had some non-black background, as most North and South American origin blacks do, owing to slavery and the resulting sexual abuse of female slaves by white slaveowners.
Vanessa Williams was announced as the first "black" Miss USA winner in 1984. She has blue eyes, which are recessive even among white people.
My personal view, based on my own experience, is that we have to go by how people self-identify. But there are some complications, again, between the US and the UK. It is in my experience true that in the UK we are more likely to use the term "mixed-race' where we Americans assume "black." (I use we in both cases because I lived in both places!)
Here in the UK I have a good friend from the subcontinent, Sri Lanka to be clear, who refers to himself as "black" because he's very dark-skinned. But in North America "black" definitely means of African ancestry. And really, the subcontinent seems to me where the question really arises: what is race? is it skin colour or what? Because if your ancestry from the southern bits, you will tend to be much darker; if you're from the north, you tend to be much lighter, owing to the fact that northern India was the home of the original Aryan "race." (Google it!)
Americans haven't yet looked at Kamala Harris's other side because there aren't as many south Asians in North America as in the UK, so they don't really have a mental category. But look at the tangle they still sometimes get into over the definition of "Hispanic": most forms there now offer categories like "Hispanic White" "Hispanic Black" "Hispanic other". In fact, official US census forms do not offer a specific category for "south Asian" but do not only for "East Asian" but in fact for subcategories such as Hmong, etc etc.
In short, I have no answers. But like OP, I think it's a valid point of discussion, as long as no one around the table starts getting upset. Like other PP, I wish I didn't constantly confront this question on forms, but I'd be glad to have more discussion over my dinner table. Then again, I got quite upset recently when some of my dinner guests said that Middle Easterners are not white my reactions was, Arabs are clearly not Anglo-Saxon but they are clearly white? Do Brits think the only kind of "white" there is is Anglo-Saxon? I also get a bit worked up when filling in forms for my children they are even more white than I and basically look white, but, still, they're my children and people in the UK, though not quite sure and usually polite, can definitely tell I'm not... Anglo-Saxon. I don't blame anyone for discussing it, though: I'm glad we're discussing it.