is chicago so cold because of the lake effect?

The "lake effect" is a word used only in the US. Elsewhere it is known as advection fog or advection snow.

For this to happen, you need wind that blows mild and moist air from a large body of water, over a cold land.

During the fall, when lake Michigan is still warm, if the wind blows from the east, that air will move toward Chicago, cool down and form fog, drizzle and even snow if the temperature is below freezing point.

The same happens here where I live, near the Oslo Fjord, in Norway.

Is lake Michigan influencing Chicago? Absolutely! such a large body of water is regulating the temperature, much like any maritime climate: milder winters and cooler summers.

But at our latitudes, the weather comes mostly from the west and I don't know enough about your country but I should think that the easterly shore of lake Michigan is more affected than Chicago.

I know that Chicago is called the windy city but I don't think it has something to do with the lake that, in itself, can't generate much wind. But the fact that Illinois is relatively flat and the city has many skyscrapers, it wouldn't be suprised of a gentle breeze feels like a gale wind in the streets between the tall buildings. And if the air is moist, from the lake, it surely must feel pretty cold.

Lake Michigan impacts the climate of Chicago and maximum of northeastern Illinois. the large thermal mass of the lake has a tendency to average temperatures, inflicting cooler summers and warmer winters. between the biggest advantages is cool lake breezes that grant some alleviation from summer season warmth

I live near the lake and often travel to the south-suburbs.

Yes, the closer to the lake you are the "cooler" it is, this can vary, today it was about 10 degrees warmer in the burbs than near the lake.