Is skyr supposed to be fat free?
Yes, Skyr has only 0.2-0.5% fat, as it’s traditionally made from low-fat or skimmed milk.

We tend to perceive Skyr as a yogurt variety, but technically it’s a cheese: Like yogurt it’s made by adding either Skyr from a previous batch or a mix of bacteria to raw (traditional) or pasteurized (modern concession to food safety) milk and then cultured, but unlike yogurt traditional recipes also use a bit of rennet. This detail may be relevant for vegetarians, depending on the kind of rennet used.

If you are missing the creaminess of 6%-fat yogurt, note that Skyr is typically served with fruit, milk and often sweetened. You could add a splash of cream instead and see if that’s more to your liking.