The only reason anyone cares about Shinola is that it is part of the slang expression not to know shit from Shinola 'to be completely ignorant'. Presumably this was inspired chiefly by the alliteration of the two words, but it's notable that the -ola suffix was common in scatological jokes in the 1930s. The expression takes on a further irony in modern times because most people really don't know shit from Shinola; they've never heard of Shinola.

It's hard to date the expression exactly, as with many vulgarisms; the first clear example is from the early 1960s, but the euphemistic variant "not to know sugar from Shinola" is found in the early 1950s, and a very reliable source, the linguist Raven I. McDavid, Jr., has quoted a limerick using the full phrase that he heard during the 1930s.