I had a student ask me about the following. I began to confuse myself to be honest. I know that we can use the phrase "beef heart." But what about when talk about sheep heart - do I ask my butcher for sheep heart or mutton heart? And what about pig heart/pork heart? Is there a good reason to use one or the other? I thought perhaps because a pig has a heart but a pork does not have a heart, so pig heart is correct. Pork can't have a heart since it is the general term for all pig meat. What about other animal hearts? chicken - I know we can use chicken heart. We would not say poultry heart. But what about duck or goose? Any ideas are helpful as this comes from a student who lives in a country that often eats organ meats. So going to a butcher and asking for sheep's liver is commonplace. Thank you.
In the UK, at least, the traditional names for meats, e.g. 'pork' for pig meat, 'beef' for cow meat, 'mutton/lamb' for sheep meat are not usually used for named parts or organs so pig kidneys or cheeks or trotters, ox/pig/lamb liver, sheep heart, cow heel. You can still buy bull pizzles but I think they are given to dogs in modern times. Also duck and goose liver (usually in pâté) and you can buy tubs of chicken livers. Often a possessive form is seen, e.g. pig's liver.
The only exception I can think of is that bovine hearts are called 'beef hearts' for eating by humans, although I have seen 'cow hearts' sold for dissection by biology students.
Also, in the UK, many butchers will deny that they sell 'mutton' (older sheep meat), asserting that all of their sheep meat is 'lamb' (from younger animals).
Some unusual organs have strange names that might be to disguise their origin, or at least soften the shock... lamb 'pluck' is the heart, lungs, and liver, with sometimes the trachea added. Sheep, lamb, cow, calf or pig 'sweetbreads' are the pancreas and thymus glands, sometimes called 'heart' and 'throat' sweetbreads respectively. Brains are just, well, brains. My mother told me poorer people in London used to buy horse or cow 'liver and lights' (liver and lungs) up to about 100 years ago.
Eating offal has largely fallen from favour in the UK, so while farmers and butchers may still use certain terminology like 'beef heart', most people would not be familiar with such terms. Even my 80-year old father, who grew up on a farm and likely remembers when offal was more frequently eaten, told me a story only recently about being served a "cow's heart" while in a French hospital. The only offal still commonly eaten is ox's kidney, usually cooked and served along with beef steak, but even then it is usually just called 'kidney' and I'd bet most people who eat it don't even realise it is from an ox. There are probably other exceptions, too - some farming communities take pride in maintaining old traditions in food and I've seen some unusual things on menus whilst travelling around the country.
I would say the best way to ask for something specific like a sheep's liver would be to use the name of the animal, not the meat. Most good butchers will be able to provide anything and are quite used to the occasional unusual request.