Maybe I shouldn't walk into this, but I suppose better that I do it than someone else. So here it goes. The word donkey isn't used very often as a word of insult in American English. However, synonyms for donkey do include those that are used as insults. I won't list all of the synonyms here, but you can see a list here. Keep in mind that at least one or two of them could be considered vulgar or offensive, so I advise you to use them with caution. To be safe, I'd stick with donkey.
Also, the donkey is the animal used to represent the Democratic Party in the United States; an elephant represents the Republican Party. Nevertheless, politics is civil enough in America that few, if any, resort to name calling with use of the words "donkey" or "elephant," although you may see a political cartoon used with those animals from time to time and depending on the political leanings of the cartoonist, an insult may be, indeed, be intended.
Interesting, I would also add that "eşec" (pronounced exactly the same) means "failure" in Romanian and can also be used as an insult like "You're a failure". It most likely has been borrowed from Turkish and the meaning of "donkey" disappeared over time and became "failure".
If someone called me a donkey in any language, I don't think I would be too happy, Turkish included :D It is imaginable that someone would teasingly call someone this, but it definitely is not a normal nickname. You might be hearing derivations of "eş" though, which means "spouse"