The author does not have to be a commoner to have sympathies for the plights of the less fortunate; that is reverse snobbery. There is also considerable knowledge of the workings of English and European courts in the plays, and several working class characters are not treated very sympathetically.
The scenes described are by no means unique to Warwickshire. In fact, there are many parts of England that could have produced the scenes mentioned in A Midsummer Night's Dream and Macbeth.
Also, just because the author clearly has a strong understanding of the way normal people speak and a comprehension of some of the sleazier corners of Elizabethan life does not mean he was not an aristocrat or noble. It is perfectly possible to be of relatively high social standing but still be familiar with the working classes. The author clearly also has knowledge that a nobleman had infinitely more access to, such as travel experience, falconry, law, politics, history, medicine, and source texts that had not yet been translated into English.