Does SongADAO have a token?

The short answer is no, there is no separate token for SongADAO. You buy a Song A Day NFT, with new ones being created and auctioned every day on https://songaday.word, and then you can become a member of the DAO.

The different membership types are explained here:

There is an official $SONG token, but it is only for the NFTX liquidity pool, which you can read more about here: