Both the (prominent) arch and the shoulder squeeze during benchpress are forms that are most common in powerlifting. For bodybuilding-style or strength endurance training, they are unnecessary or even counterproductive.
The arch
The reason for making an arch is two-fold: First, it prevents shoulder impingement as the arch ensures that the angle between your locked-out arms and your upper body is below 90°. Second, it reduces the range of motion as it raises your chest, thus allowing you to lift more weight (at the expense of an optimal training stimulus).
When doing push ups (in the common form), your hands are usually around the same height as your chest, so when you are fully extended, your upper body and your arms form an angle less than 90°, which essentially replaces the need of an arched lower back.
The squeezed shoulders
Again, this form has two purposes useful mainly for powerlifting: It further reduces the range of motion just like the arch. Plus, it provides further stability - once you press several times your body weight for a single repetition, you need all the stability you can get.
When doing pushups, you automatically let your upper body hang through as you get closer to failure, which is essentially the same as pinching shoulders. The reason that no one is recommending doing so? It's probably because bench pressing is commonly seen as a competitive movement - you want to move as much weight as efficient (i.e. reduce the range of motion) as possible while still adhering to the rules; while pushups are usually utilized as a functional exercise - you want to maximise the training effect, reducing ROM would be counterproductive.
TL;DR both the arch and the 'shoulders back and down' are powerlifting techniques that are inteded to allow you to lift bigger plates at the expense of a training effect.
