Why does bread flour have lower protein content than all-purpose flour? I decided that after six months of breadbaking, I'm ready to check flours. Up until now, I just bought the bread flour in the local shop (ok, the local shop is out of breadflour and yeast, because of covid, so the sourdough is a real saver now:)). Google tells me the most important difference between flour is the protein content. High protein for bread, low protein for cakes and in-between of all-purpose. Now this is a bit strange for me already, since I guess there are three important proteins that are in the flour: gliadin, glutenin and amylase, so this seems a bit generic, but the sources seem to imply that protein content is equivalent to how much gluten will form.

So I checked the labels of the local brand I use and was very surprised that the all-purpose flour (9.8 g) had in fact higher protein content than the bread flour (8.6 g). This would mean that the all-purpose would make a stronger dough (testing in progress). This might be a local phenomenon (Hungary), but it is more likely I'm not understanding something.

Some of the texts mention how fine the milling is, but I couldn't find any definitive answer to how the fineness of the milling would alter the properties of the dough. I also think the ash content of the bread flour is higher, but it is not exactly clear to me from the labels.

A translation of the table on the labels:


energy
fat
of which saturated fat
carbohydrates
of which sugar
dietary fibers
protein
salt


Could somebody clear this up for me? How can the bread flour protein content be higher than the all-purpose? Is milling fineness important (I'm not sure I could tell the difference between the too, so it can't be extreme)?
I would just say that you are translating the terms too literally. In the USA, it is typicall to sell flour made from hard spring wheat under the name "bread flour", because people there traditionally enjoy bread made from it. In other countries, people have different expectations of their bread, and hard spring wheat was probably not even available prior to globalization. So it is logical that companies there formulate flour to produce the locally preferred bread texture, and sell it under a term which literally means "bread flour". This is a very common linguistic/cultural phenomenon - in different parts of the world, the local term for "cheese" without further qualifiers means a very different style of cheese. 

I have never been in Hungary, but on the Balkans, traditionally, there is a preference for soft, fluffy, white bread. So I wouldn't be surprised if this Hungarian company has formulated their bread flour to produce something in that direction.