Cheesemonger is an insulting title, used by Tywin and people with his prejudices as a way to belittle the magisters of Essos, who by their very existance undermine the strict distinction between nobleman and merchant. Magisters deal in exotic goods, fine manufactured goods (a sign of Essos’s economic and technological superiority), and complicated financial transactions that nobles don’t understand. So treating them like common peddlars like the ones in Westeros creates a sense of the social order restored.
As to why Illyrio cares, I have a theory about that…
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