My Grandma had several mugs like that. She called them her "CNR" mugs. I don't know where she got them but I remember several neighborhood families had them and the story was they came from CNR dining cars. On a sort of related note, we also used to refer to stewed prunes as "CNR strawberries". I grew up in Winnipeg where a lot of people worked for the railroad.

I think it should be called some kind of combination of chicken/hen/poultry + harmonica. A 'henmonica' maybe? Or, in Portuguese, 'gaita de frango'?

You just don't see many families of that caliber these days.