"fich" is not a thing, and fiSh is not poultry. Eggs are a poultry's egg and if you're referring to chicken eggs, then they are also chickens. Some REAL examples of poultry are chicken, turkey, and pheasant.
Well, if the mother were a dachshound and a lab (i know unlikely, but just go w/ it) and the dad was a terrier, then, yeah. I would consider that to be a three breed dog.