Edu is right. Dairy and poultry is a Rabbinical prohibition. While no Meat and Dairy is a Torah prohibition. Rabbinical prohibition being completely valid, as the Torah states that people must follow the Rabbinical rulings.
Dairy and poultry cause the trouble that people (both Jews and Gentiles) may be confused over what goes with milk and what doesn't. Or gentiles may mock at seeing a religious Jew having poultry and dairy, saying how he is breaking the rules of no dairy with meat. By contrast, a Jew may think because dairy and poultry are ok, then meat and dairy can't be bad. Thus, to end the confusion, all meat and poultry must not go with dairy products, nor should there be any benefit from such a mixture.
So no creamy chicken Alfredo. No creamy mashed potatoes with fried chicken.
Regarding fish and dairy, it does depend on the custom. Personally, I have never met anyone who does not mix the two. I think there are rules that make it ok (such as not mixing fish with milk... but just having fish with "dairy products"-which contain milk. In other words, technically not really eating fish with actual milk). If it were actually "no fish with dairy" there would be no beloved tradition of bagels, cream cheese and lox.
Speaking of prohibitions, no fish and meat. I never did them together because I thought it was nasty to have meat and fish at the same time. But I had no idea it was an actual prohibition until a few months ago.