I don't understand how molar mass could affect the polarity the least
One, that's not exactly what the question asked - the question was asked in the opposite direction: polarity is related to which property the least.
Two, molar mass is simply the arithmetic sum of the constituent atoms and their atomic mass (which is a weighted average of isotopes, but that's a conversation for another post). Being just a numerical sum, it has little, if anything to do with any physical property, except mass (and perhaps density, but I'm not familiar with everything that goes into density.)
How would you suggest molecular mass relates to polarity? The mass of insulin which I linked to is some 5,000 gmol
-1. Proteins are usually only soluble in water (at least only soluble and able to adopt their active conformation). If they weren't soluble in water, our cells would die from the clogging of precipitated proteins, and our bodies would have to find a new way to function fast.
All I'm trying to say is molar mass is not a valid indicator of polarity - or to rephrase to fit in line with the wording of the question - polarity is related to molar mass the least.