Guys, remember your Archer. Idioms are just colloquial metaphors. They're expressions used to describe an attribute. Some may not make any sense when translated into a different language. Not necessarily because of the words themselves, but because of the cultural meaning behind them.
this is kinda weird. Duro means hard (in both english meanings of not soft and, less often, difficult) but is hardly used for people. There is an idiom about people with duro, that's "essere duro di comprendonio", which means "to be very slow at understanding things" and it is sometimes used omitting the last part e.g. "ma sei davvero duro! Lo vuoi capire che in inglese ci sono due traduzioni per l'italiano tempo? Time e weather! Non sono mica la stessa cosa." But duro as stubborn is weird. This should be "testardo come un mulo" (which is btw an idiom you can hear quite often)
Well, just because in English we think of mules being stubborn, there's no reason that Italians should. It seems likely to me that duro is used in the same sense here as it is in the hard of understanding phrase you quoted. So mules are stupid in Italian and stubborn in English. That's how idioms work.
I understand this is idiomatic. But in English you would have to say "as something as something else". You need the "as" repeated to make it work. So "as stubborn as a mule"because the whole point is to compare one thing with another. My tablet version of the app only gave me one "as" tile. So I have to use a clumsy phrase you wouldnever use in English to get the points!
I encountered the same issue on my tablet app and had the same thought at first. However, I think the first 'as' is regionally dropped and not absolutely necessary, at least in US English. I've heard the phrase used both with and without that first 'as.' Also, if you consider 'as' synonymous with 'like,' and "You are stubborn like a mule" makes sense, then "You are stubborn as a mule" is reasonable.
Allow me to add a quote from Wikipedia: "A male donkey or ass is called a jack, a female a jenny or jennet; a young donkey is a foal. Jack donkeys are often used to mate with female horses to produce mules; the biological "reciprocal" of a mule, from a stallion and jenny as its parents instead, is called a hinny." Ass and mule are not synonymous.