Questions about which social class one may be are usually asked by the middle class or those aspiring to be middle class. The upper class (aristocracy) know what they are, and the 'underclass' (a newer term which describes an age old concept) know what they are.
In British history, there has traditionally been a division between the aristocracy and the rest (those whose service or lands are owned by the aristocracy). Then, somewhere between William the 1st and Queen Victoria, the merchant classes and middle classes (a troublesome lot) began to challenge the order a little. A farmer called Olly really rocked the boat in the 17th century, indeed the Civil War was a class war, and although we got a republic, we found Olly a bit too big for his boots an no-one really liked him, so we get a real king back on the throne. You see, there is something about having a monarch and an aristocracy that keeps everyone else in their place
With Victoria, the working classes (very useful in the Industrial Revolution and as cannon fodder for the Empire) began to gain some nominal power, and some autonomy (what with Chartism and all that). They even began owning houses and attending classes on how to be clever.
The 20th and 21st centuries have brought greater spending power to all classes (except the underclass) and legislation makes it illegal for us not to be equal, but 'class' is, funnily enough, not a protected characteristic (and neither is accent which some people assume is an indicator of 'class').
So, back to the question of 'What the hell is middle class'? I would suggest that today, it is not about what your daddy does (an old measure of class) nor the pedigree if public school educated ancestors. It is all abut money and how rich you are, just as it was when the merchant classes rose.