Translation:I also eat fish.
You are correct: 我也吃鱼 can indeed mean "I, too, eat fish" (here, "too" refers to "I") or "I eat fish, too" (here, "too" refers to "fish") and there are at least a couple other possibilities: C) "What do I do when I visit Florida? I swim in the ocean. I collect seashells on the beach. Also, I eat fish." (Here, "also" refers to "eat.") D) "I study fish. I catch fish. I also eat fish." (Here, again, "also" refers to "eat," but specifically within the scope of the writer's interest in fish in particular. In English, we can suggest this meaning by putting "also" at the beginning of the sentence, or more effectively in speech by emphasizing the verb: "I EAT fish, too.")
In Chinese, if you want to specify your example "A," you could provide the context that clarifies the meaning, e.g., 我的猫吃鱼。我也吃鱼。(My cat eats fish. I, too, eat fish.) or, you could use the particle 啊 (pronounced "a") to emphasize 我 : 我啊也吃鱼。Using 啊 this way means something like "Me, I also eat fish." or "As for me, I, too, eat fish."
As for your example "B," again, you could provide context: 我吃米饭。我也吃鱼。"I eat rice. I also eat fish." I have heard that the particle 了 (pronounced "le") can be used to intensify a preceding clause; however, I do not think that option would work to emphasize "fish," inasmuch as placing 了 at the end of a sentence usually indicates past action - the Chinese equivalent of putting the sentence in past tense: 我也吃鱼了means "I ate fish, too" (not "I eat fish, too"). So, I still do not know how you could specify the meaning of your example "B" without context.
These are great examples Seal! I would like to add that a sentence-final 了 does not necessarily make the sentence past tense. It can also indicate a new situation. For instance, maybe you were a picky eater as a child, but now you eat a wide variety of foods. You might explain to your childhood friend: 我也吃鱼了, meaning "I also eat fish (and I didn't before)."
It means that if the person this is said in response to was already speaking in the past tense. Tense markers are not necessary if the tense is understood from context. If the person says I am eating fish, or later I will eat fish, you will need to use 了 to clarify that you mean in the past. In this case 我也吃鱼了。 would be needed