It's even more inconsistent than that. Gairaigo is sometimes used for meat. Apparently, the animal is still typically referred to with a non-gairaigo word, but the meat is sometimes (but not always) referred to by a gairaigo word.
- ラム is sometimes used for lamb (the meat)
- マトン is sometimes used for mutton (meat of an old sheep, hence the English expression mutton dressed as lamb)
- チキン is sometimes used for chicken (the meat)
- オージービーフ for Aussie beef
This inconsistency isn't unique to Japanese, though. In English, the words for animals come from different languages than the words for the meat: the animals are of Germanic etymology, whereas the meats are of French etymology, according to this answer on Linguistics Stack Exchange. The Germanic words would be older than the French words.