In algebraic $K$-theory, $K_0$ and $K_1$ have nice descriptions in terms of the category of finitely generated projectives. $K_0$ is motivated as the "universal receptacle" for (additive) invariants of projective modules ("Euler characteristics"), and $K_1$ as the receptacle for invariants of their automorphisms (as in these notes Def 7.4). There is the paper of Blumberg-Gepner-Tabuada. Also, at a comment on MO (2nd answer) it is suggested there should be a description of $K_2$ in a similar vein.
Question. Is $K_2$ in a similar way the "universal receptacle" for additive invariants of something to do with projective modules? ( ...and higher $K_n$?)
Perhaps I should say: I'm thinking only of commutative rings here, and $K_2$ in the sense of Quillen (or equivalent).
Perhaps the result in my paper on multidimensional acyclic binary complexes at https://faculty.math.illinois.edu/~dan/cv.xhtml#binary will address your question, even though it doesn't treat projective modules in a special way.