In this case "fiskens kød" literally does mean "the fish's meat" and won't be used for saying "the fish meat", which would instead be "fiskekødet".
With this sentence, "Manden spiser fiskens kød", it could both mean that the man is literally eating the fish's meat, as in meat that belongs to the fish, and not the fish's own flesh 'and blood'.
So like you pointed out, "fiskens kød" is "some meat that is possessed by the fish".
Now, "fiskens kød" could also refer to the meat on the fish. As in, "Spiser du fiskens kød?" = [Literally: "Eat you the fish's meat?"] = (Meaning: Are you eating the fish's meat? / Do you eat the fish's meat?). However, as a native Danish speaker, I can't see a real-life situation where you'd say this.
Unless you got very technical, and you were two people talking about which parts of a fish you would be eating:
Person 1: "Spiser du fiskens kød?"
Person 2: "Nej, jeg spiser fiskens øjne."
Person 1: "Do you eat the fish's meat?"
Person 2: "No, I eat the fish's eyes."
But unless you wanted to get specific like that it seems a bit redundant. If you wanted to say, "Do you have the fish meat?", you'd say, "Har du fiskekødet?" = [Literally: "Have you the fish meat?"].
I hope that helps!