Is a nonsense verse actually intended to be serious?

This is a personal opinion: Every verse including nonsense verse says something about the writer, and in that sense can be serious to some extent. Sometimes the verse itself may have a 'moral' or central theme that serves to instruct. Carroll's material in Through the Looking Glass are good examples. There are lessons in "The Walrus and the Carpenter", and even in "The Jabberwocky". But there is no requirement that the author must set out to be serious when writing nonsense verse.