I haven't read the poem, I have to admit, but I still believe that I might be able to say something about what the title might suggest.
To me, the title seems to go against the popular notion that all women share basic things in common and thus are naturally friends, allies, or "sisters." This popular notion is often called, often disparagingly, "essentialism" or "radical feminisim." The reference to the moon in the title is telling, I think, as it evokes the whole notion of the 28-day (or so) lunar cycle; the moon often functions in literature as a symbol of the feminine, probably in part because the lunar cycle seems to coincide with the menstrual cycle for many women. ("Coincide" in the sense that the average menstrual cycle tends to be about 28 days, not in the sense that the moon is in any way involved in the menstrual cycle. "Under the same moon," now that I think about it, may even be hinting at the idea of women's cycles 'synching up" if they're around each other a lot.)
Critics of radical feminism often point to how women are not a single, unified group. There are profound differences of race, class, sexuality, nationality, ethnicity, etc. The idea that all women are "sisters" may be appealing to some, but others see it as a myth that obscures the real differences in power and status that different women experience.
The poem "We're not all sisters under the same moon" refers to social feminism and sisterhood. Social feminism stands for women's differences and their struggle against patriarchy.
The poem suggests that women are not the same but different and they don't have the same enemy. The moon is associated with woomanhood. As the moon turns into different shapes, women are different because of their cultural, historical and social background.
Women have to acknowledge that since they're different there' is no such a thing as Sisterhood. Only then they could find the way to communicate.
Jackie Kay talks directly to white women (when you see my tone changes... )and asks from them not to give black women a definition as white women because there are a lot of things to separete them. Black women have other problems.
She hopes that her words will make the woman wordwide to understand and accept their differences and the fact that there is not only one way to fight against patriarchy. Feminism as we know it is not the right way to accomplish our goals. Women have to rethink about their strategies.