Doesn't know shit from Shinola
Possessing poor judgment or knowledge.
Shinola was a brand of shoe polish previously manufactured in the USA. The alliteration and the fact that the two commodities in the phrase could possibly be confused is the derivation. The distinction is well made; only one of them would be good to apply to your shoes and only particularly dim people could be expected to muddle them up. Of course, outside America, most people don't know Shinola from anything at all, as they've never heard of it. Even in America it would probably not be widely remembered but for this phrase.
The 'ola' suffix is popular in the USA as part of trade names, e.g. Crayola, Granola etc. This leads to the pronunciation of Shinola as shine + ola. That spoils the alliteration a little as it would work better as shin + ola.
This phrase is typical of the barrack room vulgarity of WWII, which is where it originated. Other "doesn't know" phrases, also mostly from the military are, "doesn't know his arse from a hole in the ground" (or elbow, or a hot rock, or third base), "doesn't know enough to pee downwind", "doesn't know whether to scratch his watch or wind his ass". The tone is lifted a little by the English conductor Sir Henry Wood who expressed a similar opinion with "he doesn't know his brass from his woodwind".