Holding weight overhead without arching your back demands a lot of mobility that many of have lost. This position demands even more range of motion when holding the weight while simultaneously squatting. For the athletes you see below shoulder mobility was of highest priority in order to get that tremendous amount of weight overhead.
To look at it from the other side, and more bluntly, if you cannot put your hands over you head while keeping your chest down then how could you ever do the same thing with weight in your hands?
Because the athlete pictured above lacks complete shoulder range of motion he must arch his back in order to put his hands over his head. If he was to press weight above him in the position shown his lower back would take the brunt of the load and his spine would likely get injured over time. The only way he can avoid hurting his back when pressing is to keep his chest down and abs tight. However, he would have to essentially press the weight at a 45 degree angle in front of him in order to do that. With any significant amount of weight that is going to be impossible.
If you want to check your shoulder mobility here is an easy test. Lay on the ground flat on your back. Now raise your hands directly overhead until your hands touch the ground with your elbows straight. Did your lower back raise off of the ground? If so or if it is really hard to make both your lower back and hands touch then you need to work on your shoulder mobility before you lift weights overhead.
While you are improving the mobility of your shoulders you can still press, just not directly overhead. Try incorporating movements that purposefully use the angles where you can keep your abs engaged. Include movements like bench press, incline bench press, and landmine presses into your routine. Each of these movements allow you to keep your back from arching while pressing with similar muscles as the overhead press.
Always train and strength the ranges of motion you currently have. Separately train your body’s flexibility so you will be able to incorporate even more movements and muscles in the future.