How to achieve this look?

I've been searching for this exact thing! Oil filter gets close, but not quite!

Wanted to mention that the effet you are looking to use only looks good at large scale size. For example on most consumer packaging design we can never use on for example a candy wrapper, but on a large pet food bag yes. Basically the brain/eyes will not register the effect unless is large enough to see the brush details.

A vector file with this much detail, to keep the realistic look, might be a bit complex for most systems. As Michael mentioned, the Oil Paint Filter in Photoshop would be a possible solution—as well as the Live Oil Paint Brushes in Fresco or by using Substance Painter and Substance Designer.

Does this have to be done in Illustrator? Much of this look reminds me of the Oil Paint filter in Photoshop.