In the example, I think what Feynman is describing, we commonly call "visualization", to be able to "see" the problem in imagination. That is no less a form of abstraction that is often vital to problem-solving. Of course, not every problem yields to this approach but it is a powerful feature of our basic cognitive tool set.
Einstein wrote that his early success in formulating his idea of special relativity was the outcome of his intuition about the physical properties of light, etc. he studied. Later on, the mathematical abstractions became more powerful, and at that point intuition about the "physical" nature of phenomena was insufficient for understanding.
But I think there are forms of intuition that apply to very abstract ideas, or what seem to be so to us. I once heard a physicist say "we never really understand higher mathematics, we just get used to it". Feynman would probably have agreed with that sentiment.
"Getting used to it" is really the equivalent of developing an intuition about the subject. I remember first learning about programming recursive functions, mind boggling in the beginning. After a while, it began to "sink in", that is, it became intuitive, I no longer had trouble "seeing" how it worked. The key is familiarity, something once strange is now digestible.
So there's nothing binary about intuition, it covers many forms of thought, and incorporates reasoning about emotion, having a "feel" for the problem in question. There are limits to our abilities, at the highest level it's genius, but there are no clear boundaries.