The halacha of a human being is considered separate from the halachos of animals. Thus, animal milk from a kosher animal is dairy, while milk that is not kosher would not have the extra restrictions of dairy (since it is not kosher anyway). However, human beings are not considered in the laws of animal food. It appears that the definition of dairy is milk from a kosher animal, which does not apply to a human being. The main halachos of this are mesorah from
לא תבשל גדי בחלב אמו
Breast Milk: Just Like Honey From a Bee?
By way of explanation, one contemporary rabbi compares mother’s milk
to honey—as a similarly exceptional case of kosher food derived from a
non-kosher animal.
That is, bee honey is explicitly shown as kosher, since Yonasan ate it. Breast milk is explicitly shown as being given to Yitzchak, Moshe, and Shmuel which implicitly shows that is is kosher in that the Tanach references it as a normal food. This is the case even though the source is not a kosher animal. We also see that an infant should not nurse from a woman who has eaten non-kosher food if it can be avoided.
Shulchan Aruch YD 81:7
Thus, Is Breast Milk Pareve? points out that it is pareve, while not giving the specific reason. However, the cooking of meat in it is asur because of mar'is ayin.
It is forbidden to cook meat in breast milk (Shulchan Aruch YD 87:4). Your baby
need not wait between eating meat and nursing.
While Rambam Ma'achalot Assurot - 3:2 says it is permitted, he does not go into the reason it is pareve.
Human milk is permitted to be eaten,3 although the meat of
a human is forbidden to be eaten. We have already
explained4 that it is forbidden by virtue of a positive
commandment.5
3. I.e., even by an adult. Note, however, Halachah 4.
4. Chapter 2, Halachah 3.
5. Thus it does not contradict the general principle
mentioned in the previous halachah.
Mother’s Milk and Kosher Laws
Additionally we don’t find (other than the story of Moses – which I’ll
mention soon) that there is a prohibition to drink milk from a
non-Jew. Milk is a separate entity in itself, which is permitted by
the Torah when coming from a human female, whether Jewish or
non-Jewish. It can be compared to honey which is not considered
non-kosher, even though it’s coming from a non-Kosher insect, the bee.
However, on a more esoteric level, our Rabbis tell us that what a
person eats affects his soul. We know that the quality of a mother’s
milk is influenced by what she eats; therefore it follows that if she
ingested non-Kosher food, traces of that would come into the milk.
Therefore, although in terms of Jewish law there is nothing wrong with
giving the baby donor-milk from a breast milk bank, if it would be
convenient, and there would be a donor available who keeps Kosher who
would be happy to pump some milk for the baby (e.g. a sister or close
friend), this would be preferable.