They are both similar to what the result would be if you took cottage cheese and blended it.
I'm interested in any differences in the contents. I read the Wiki articles I linked to. In Britain skyr is labelled as yoghurt (at least the Arla brand is) and must therefore contain both Lactobacillus bulgaricus and Streptococcus thermophilus. Wiki mentions both in their skyr article and neither in their quark article, where they mention lactococcus instead. But I don't take Wiki as canon and I wouldn't assume that these and only these bacteria are included.