Whataever physiological connection exists there, is far from obvious. And I haven’t found concrete studies to explain it.
They're mainly talking about severe cases, but it doesn't seem that unlikely that it could also have more mild effects to those organs.
More recently, I've seen some scientific back-and-forth about whether Long COVID is novel or just a form of already-known post-viral syndrome. I sort of feel like it doesn't matter. Even if it is "just" post-viral syndrome, it seems to be particularly severe and widespread. Whichever theory turns out to be correct (including the "just depression" theory), the empirical fact of Long COVID's existence has more to do with my continued caution than any fear of acute COVID. Getting older has been and will continue to be "fun" enough without that.
Nonetheless, the two qualifiers in the statement "People who had severe illness with COVID-19 might experience organ damage affecting the heart, kidneys, skin and brain" weakens the argument. If the long-covid issue was just that people who were in ICU then had long-term symptoms, that's an easy premise to understand. But IMHO, for mild symptoms, there's a non-obvious leap to say that long-term brain damage somehow occurred.