I don't understand what kingdom this mushroom is in
A mushroom is in kingdom fungi. You can know this by using classification. The same way flowers are in kingdom plantae.
I also heard something about binomial nomenclature?
it's the formal naming system scientists use
Clade is also a term that needs to be known, it is a group of organisms that is believed to have evolved from a common ancestor. Like certain types of mushrooms are from the same clade. A cladogram is used for this reason.
There are five kingdoms, animal, plant, fungi, protist, fungi, and monera. Taxonomy is the science of classification.