Why must we abstain from meat but not from fish?
In the original Latin text of Pope Paul VI’s apostolic constitution Paenitemini we find the following:
Abstinentiae lex vetat carne vesei, non autem ovis, lacticiniis et quibuslibet condimentis etiam ex adipe animalium [III:1].
(Translation): The law of abstinence forbids the use of meat, but not of eggs, the products of milk, or condiments made of animal fat [III:1].
The word used for meat is carnis. In modern English, the word meat can refer to any animal flesh. But, in the Latin cited here, it refers to mammals and birds only. So, fish, seafood, amphibians, and insects are permitted (assuming you want to eat some of them); but mammals and fowls are not.