"As stubborn as a mule" is more common. I would go so far as to say that the saying _is_ "stubborn as a mule." We can do quick checks on questions like this with Google searches. A Google Books search yields 30,700 hits for "stubborn as a mule" and only 1,530 for "stubborn as a donkey." Doing some searches just for fun, I also find some rare occurrences of "He's as stubborn as a horse with glanders," "stubborn as an ass," "stubborn as a jackass," "stubborn as a rock," "She's stubborn as a jellyfish and isn't afraid to shoot her mouth off to anyone," "stubborn as a bull," "stubborn as a tree stump," "stubborn as oak," "stubborn as steel," "stubborn as iron," and "stubborn as adamant."

Agreed. Stubborn as a mule.

In my experience, "stubborn as a mule" is more common.

I don't think I have ever heard it used as 'donkey', despite the fact that mules are practically unknown in the UK these days! They will survive forever as examples of stubbornness.

this is the straw that broke the camel's back