I think what is confusing is that sometimes, "when "of" is used in English, it is correct to translate it to "av". But here it is not correct. I think that there are actually many variations in how we use the word "of" which native English speakers don't consciously differentiate. Unfortunately, I am a native speaker of English too, so I don't know what they are either, although I want to know!
English does tend to use "x of y" construction quite freely. For Swedish, there are usually two basic cases where it translates directly:
- Titles are the exceptions to possessives, so "the King of Sweden" = Kungen av Sverige.
- Partitive expressions, meaning that x is a quantifiable part of y, use the same construction: "one of the dogs" = en av hundarna
The latter occasionally extends to meronomy, meaning that a part of a whole is expressed in this way ("the wing of the plane"), and it can be tricky to figure out when. More often than not, however, we express this differently.
I was thinking more from the english perspective as I have always figured beer and ale to be essentially the same and sort of like a dialectal thing but I suppose from a technical standpoint they might not be. Also the bab.la and google translate list öl as an alternate spelling for ale (although going off of what you have said maybe that is no longer common?)