We put categories on things that we see outside in the world. We say an explosion is an event, a happening in time and space. We say stuff is made out of matter. Yet we all know that emotions exist too. Are they events in time and space? If yes should they be categorised as events? Could they be classified as matter? If yes, is it a specific structure of matter which is common among everyone experiencing that emotion, say sadness? Would intuition be categorised as a structure of matter in the necessary context? Or should we really give emotion ontological status?
Emotions are a feature of our consciousness, so this question is basically asking what the ontology of consciousness is. That is the central question of the Philosphy of Mind, and it has been a focus of much philosophic thought for the last century. This is because, as your questions note, the dominant movement in philosophy over the last century, physicalism, struggles to answer what consciousness is.
Most physicalists assume some version of Identity Theory, and if one does that, then emotions are a reflection of physics, and are themselves causally irrelevant, as are all conscious events. Property dualism gives some additional ontology to emotions, and awareness, but in most forms (dual aspect theory) it has similar causal irrelevance, in that the real causal work is done by the physical aspect of matter.
Substance dualism asserts there are mental and physical substances in the universe, and emotions are a mental substance, which our minds host. Substance dualism involves an interface principle, in which the mater of our brains can affect the thoughts and emotions in our minds, and vice versa.
A "physicalist" version of this is a strong emergence theory, that mind emerges from some unique matter structures, but is then at least partially causally independent of its substrate.
Physicalism has a similar problem with ideas (math, ethics, Beauty, logic, causation, time, etc), in that they seem to be important in the universe, but don't seem to have any role in physicalism. Frege, and the Popper, therefore proposed that the ontology of our universe is triplest -- IE, matter, mind, and ideas all have their own existence. Popper combined his triplism with strong emergence for mind. https://tannerlectures.utah.edu/_documents/a-to-z/p/popper80.pdf