The short answer is yes. Or no, depending on what you mean. A lot of different people have analyzed folk music, and or folk songs, but they use differing methods and differing analytical tools, depending on what they are looking for.

It is certainly easy to analyze the lyrical structure of the verse, if that is what you want to do, using the names like dactylic hexameter, and to number the lines, and using names for line/verse structure that are used in poetic analysis, like Petrarchan or Quantrain, or whatever else-- Different verse structures and patterns of accent have different names, often based on the poet who used them, or the time and place that they were popular--

There is also a language for the structural analysis of music--since folk music generally is musically fairly straightforward, structurally (it tends to follows the verse structure) and, as has been pointed out many times here, there are many songs but relatively few melodies, people who analyze the music are usually more interested in the histories, evolution,derivations and transmission processes, than of the purely structural aspects of the music.

How you analyze things really depends on what your objective is--the basic analyses correspond to the basic elements, which are scale and meter for melody and the rules of verse for the lyrics-