How can I achieve this feeling of a drawing?

Yes, it can be done in Illustrator.

As already mentioned, obviously the Grain effect is used in your sample image. The effect is located in the Effect menu. Just draw some filled paths, apply the Grain effect and play a bit with it and its settings. You will see that it gets close to what you want to achieve as per your sample image.

You may also fill your shapes with gradients and use some gaussian blur or feather effect and the like. Also, various opacity settings and blending modes may help (see Transparency panel).