What's the noun for genericalness? Here's the word you're looking for.
general
(now rare) A general fact or proposition; a generality. [from 16th c.]
(military) A senior military title, originally designating the commander of an army and now a specific rank falling under field marshal (in the British army) and below general of the army or general of the air force in the US army and air forces. [from 16th c.]
(law) With respect to a trademark, the act or process of becoming so widely identified as a reference for a type of product as to cease representing a particular brand or manufacturer of that type of product.
“He finds it difficult to separate theme from plot, and to express the theme in terms of a generality rather than a specific taken from the particular story.”
“I shall try accordingly to avoid the usual generality about Bacchylides' elegance and polished style, a commonplace of Bacchylidean criticism.”
“The impressive thing about it is Abel's monumental generality in his statement of the problem.”
“When you jump up a level in intelligence, all the old problems are no longer fun because you're a smarter generalizer and you can see them as all being the same problem.”
“CELL, ERIC'S TRIP 6/15, EMPTY BOTTLE After DGC Records scooped up New York's Cell on the basis of one single, their tedious debut, Slo-Blo, proved the move premature with its indie-rock genericism.”