Common enough (though there's a diversity of pronunciation and spelling of the name), though for any practical purpose probably "edit distance" works better. (And Hamming distance would probably be better called XOR distance or something.)

Funny coincidence, there was a comment on HN the other day when someone called Euclidean distance "bird distance", and everyone agreed it was a great coinage. I think "Manhattan distance" is similarly evocative, and no less precise than any alternative.