In short, yes the wheel size does mean something.
In long, a smaller wheel with bigger tires is more prone to tire deformation than a larger wheel with less tire. When a tire is under deflated or over inflated, the tire itself changed properly. A tire with less PSI will sag down and offer more grip while driving on the sidewalls in a way. This is why offroad tires have aggressive sidewalls or even extra thread on the side or corners next to the normal patch of thread for when a vehicle goes offroad, they are often running around 15 PSI in the tires for the extra grip when going over rocks. Now if you over inflate the tires, when they heat up, the tire will expand and instead of driving with the full patch of threat on the bottom, the tire balloons and the tire will become more of a U shape, only offering the middle of the tire the thread patch. This will give less traction and can give an imbalance of thread wear which can cause unwanted problems with the tire. That is why most vehicles will be set to around 32 PSI for normal driving conditions and also why people will shoot for that magic 32 PSI while tuning a vehicle.
With this information, a car will lose grip if the tires are inadequate. But with a smaller wheel and larger tire, it might not be enough grip for the car even with running a weird PSI in their tires so to combat this, a person will use a larger wheel to give the car more grip. Think regular tire meets a low profile tire. The main benefit to this is less tire deformation. So they will maximize the wheel size to offer more tire thread to the ground which increases the grip the tires will get. If you look at your tire size range and move left to right, you can actually see the tire changing with the wheel when you look at the rear of the vehicle and look at the back tires on the ground. The larger the wheel, the more thread and flat looking the tire looks on the ground. You will also notice that the “balloon” look of the tire looks less the larger the wheel gets.
To put this into perspective, lets say on your Abarth the tire width is 250cm and the wheel size is 13. Just looking at this, it will look like the thread of the tire only reaches around 80% on the surface. So effectively, you are only getting 200cm of your tire to the ground instead of the full 250cm. Changing the wheel from 13 to say 18, so increasing in each size will roughly get you around 8.33 (repeating) in the thread that reaches the ground under normal 32PSI inflation. Which means at a wheel size of 18, and the tire width at 250, you will get the full 250cm of tire thread on the road without deformation of the wheel under harder driving conditions.
When it comes to racing, you will see this most often in rear wheel vehicles where the weight is usually more in the front than in the rear. Cars will be set up with a rear tire that is slightly larger than the front. I have even seen this personally in races such as the Pike’s Peak Up Hill Climb (Race to the clouds) where a car has ran 17’s up front, and 18’s in the rear and even saw one with 15’s in the front and 20’s in the rear to get extra grip in the rear for traction. In online races, including drifting, normal racing, playground games, and even on FM7, there are some cars that will run the larger wheel in the rear than in the front because 1) they need additional traction in the rear, and 2) they can’t put the front wheels larger to the front because of the PI index increasing even more for the front than the rear, this is due to having more traction and thus better braking and handling. On the Abarth, if you change the wheel size from a 13 to the largest setting offered, like 16 or 18, you will notice that the 0-60 time can be effected and usually reduced as it is getting more traction and has less spinning than if it had the 13’s on it. I have seen cars lost as much as .6 seconds off the times offered. Yes, the larger wheels will be heavier, but the traction gained usually outweighs the actual weight gain of the car.