There is no reason why the first font name should be quoted, but presumably the declarations that you have been reading have often mentioned a name consisting of several words as the first name. Many fonts have such names, and probably other font names that people list down have been copied from some list of common fonts with simple (one-word) names.
If you view the CSS code for SO pages, you will see font-family: Arial, Helvetica, sans-serif;
. No quotes. No quotes are actually needed around Times New Roman
either, but people tend to take warnings seriously enough and quote such names.
So unless you can provide evidence of declarations where several multi-word font names are listed and only the first one is quoted, this is just a symptom of simplistic thinking: people may consider the choice of primary font, possibly selecting a multi-word name, and then they write a more or less random list of other fonts, with simple names.