wonder what makes a song secular if it's still sung by monks?

The words, of course. "Sumer is icumen in" is most definitely a secular song, but it has survived with both words an music because it was written down by the monks of Reading Abbey. There is also a set of sacred words in Latin on the same MS.

If you go here, there is a scan of the MS which clearly shows both the English and Latin words.

I bet the monks did not only sing the Latin words on that manuscript <g>