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You guys are aware that the Chicago machine is ran by an oligarchy of rich old Irish guys, right?
Yes. The OP asked what the GOP meant using the term "Chicago-style politics".
Now, as much as I am inclined to think of the GOP as ignorant fools, there are a lot easier ways for them to imply anti-black racism than by using "Chicago style politics", a phrase which has for DECADES implied a degree of lace curtain corruption and machinery that harkens back to the days of Tammany Hall. It clearly implies a degree of corruption, not some racial code bullsh*t they could pull out of their asses using a multitude of other sayings.
The implications of "Chicago governance" has been around a lot longer than Obama has been president. And while yes, Republicans do misuse the term for anything Democrats do now days, assuming it's a racist code phrase for "blacks" is ridiculous. Even if used by Republicans.