Misrepresenting the target group is just an added bonus. It's like referring to all indigenous people as "indians" - even if they were from India (which they're not), it's just taking a blanket term and making them all seem like they're the same. At least that's why it's even funnier in this context.

But generally speaking, when people unironically use the term to refer to all whites, well, that just shows that they don't really understand the meaning of the term they are using.

Why are they referred to as the white race?

Yeah, it's a term from an outdated racial theory that got into mainstream language. Still keeps weirding me out every time I see it. In Russia where I grew up, actual people from Caucasus region are still often treated and portrayed in media as "black" for being generally slightly darker than slavs. It's disgusting. But, the populations in Caucasus region are actually very diverse. Like, for example, just talking about the skin color, the Azerbaijanis are generally fairly tan, but then quite a few of them are whiter than snow, and that's the diversity within just one ethnic group of which there are few dozens. So I find it just as weird to call white people Caucasian (and people of Caucasus just "white") as it'd be calling them Romanis or Tuaregs (or Jews) for how meaningless that'd be.