why is chicken "meat?"

The Short Answer:
Neither chicken nor fish meet the Torah 's criteria for meat (in this context). Chicken1, however, is rabbinically considered meat and forbidden with dairy.

The Askmoses Answer:

The Torah says2 "do not cook a kid in its mother's milk".

The oral tradition3 tells us that (in the Torah's classic way of conveying many messages through limited words) the Torah is teaching us many laws through this enigmatic verse. For example: "kid in its mother's milk" is not limited to a young animal in its biological mother's milk, rather the Torah is describing the general definition of meat for this law. One of the criteria for "meat" in this context is that its mother gives milk.

since people consider chicken to be a form of meat, it is now in the category of meat and may not be eaten with dairy.
This then does not include poultry or fish4. Thus technically speaking, there is no Biblical prohibition against eating chicken or fish with dairy.
However, this poses a problem: many people did, and continue to, consider chicken a form of meat. It was sold together in the market, and is sold together in butcher shops and at the deli counter of your local supermarket etc. It was/is thus easy to make the mistake that if chicken (which in the mind of the masses is meat) was being eaten with milk, that meant other meats, as long as they aren't the actual kid in its biological mother's milk, are also permitted with milk.

Obviously this is incorrect; you may not eat any "beef" (or lamb etc.) in any (animal's) milk.

This is a troublesome thought. People could innocently make the mistake and break a Torah law. The sages, who were commanded by the Torah to protect its laws, issued a decree that since people consider chicken to be a form of meat, it is in the category of meat and may not be eaten with dairy.

This confusion never applied to fish and fish is still not classified as meat. (There are other issues with fish, both regarding fish and milk and fish and meat. See each link respectively).


Footnotes
1. We are just using chicken as a common example. However, the same applies to all poultry.
2. Exodus 23:19, 34:26 and Deuteronomy 14:21
3. See Talmud tractate Chullin 113a (and subsequent pages). Also see Rashi on above verses.
4. The difference is that chicken is considered a form of meat - but it is excluded because its mother doesn't give milk; whereas fish is not even considered a form of meat in the Torah.