That's not a question that can be answered with a simple yes or no.
Fire the same weight load at the same velocity from two shotguns of the same weight and fit, the recoil will be very similar regardless of gauge.
20s use lighter ammo for the most part, but oft weigh enough less that the kick feels the same or more so. There's other factors.
The 20 gauge 870 YE here is a stout kicker with 1 oz loads. It has a 21" barrel that's the same length as the one on Frankenstein, my 12 gauge parts 870. Frank kicks little with oz loads, only partly because it's slightly heavier. It also has a larger pad and butt. This spreads out the push.
To lessen kick, one can lighten the ammo, slow it down, and/or make the gun heavier. Target shooters shoot 100 rounds a day or more in trap and skeet events. Target shotguns run heavier than field guns and use lighter loads for the most part.
New shotgunners oft take a beating because their form is bad, they use light shotguns and heavier loads than optimum. Regardless of gauge, get your form right, use light loads, and you'll have lots of fun and get proficient in short order.
HTH....