What kind of writing is this?

I would say these are musical lyrics which usually rhyme and have rhythm to make them fit music. Tolkien usually made it clear that his characters could sing songs almost spontaneously since they were so musically in tune with themselves. Of course, we don't have the music to go along with it, but then we don't the music to the psalms of the Bible which were also sung over 2,000 years ago either. This is one ability we've lost in our modern civilization to spontaneously sing like Tolkien's characters do so freely at any time of the day for relaxation, succor, and diversion. Today we turn on the radio or some other electronic devise. In these ancient times of The Lord of the Rings, all the characters provided their music, song and musical lyrics.