All resistance exercise involves forces being applied to the body in two distinct positions, with the forces acting in opposite directions. In any standing weighted exercise (squat, overhead pressing, deadlifts, etc), there is a downward force applied to the body by the weight, and an opposing upward force applied to the soles of the feet by the floor. It's normal to only think about the weight being moved in resistance training, however alternate way to consider how the exercise loads the body is to think of the trainee as being sandwiched between these two forces, which they have to resist.
The difference between pushups and bench press can then be seen as different forms of loading through a similar movement pattern. In the bench press, the arms are loaded between a posterior force on the palms and an anterior force on the shoulder blades. The shoulder are kept retracted and depressed so that the shoulder blades remain the contact point on the bench. If you were to protract the shoulders, then the spine would become the contact point on the bench instead, which would have two effects. Firstly, the weight is now being held over a single point of contact (the spine) rather than two points of contact (the shoulder blades), which is a much less stable position. Secondly, the load now must also the transmitted through the acromioclavicular joint, requiring additional muscles (serratus anterior) to support it. These two factors will reduce the amount of weight one is capable of lifting, making the bench less effective at its usual purpose of chest and arm development, and definitely less advantageous for bench press competition, if that is a goal.
In a pushup, the whole body is loaded between the posterior force from the floor and the anterior force of gravity distributed over the whole body. If you simplify the pushup forces to just the weight of the chest over the hands, the anterior force can be thought of as loaded through the ribcage. So let's look at the concerns about scapular protraction in the bench press and see if they apply to pushups.
Firstly, there was the issue of instability with a single point of contact between the shoulders against the bench. In the pushup, the anterior force is applied in front of the shoulders, rather than behind, making it inherently stable. Imagine maximal shoulder protraction in a bench press, with your feet off the ground - it would be easy to roll to the side. But in a pushup there is no lack of balance that could cause you to suddenly roll sideways, unless one arm completely gives in.
Secondly, there's the issue of involving additional joints and muscles as potential weak links in the chain. In the bench press, keeping the shoulder retracted avoids loading of the AC joint, so the AC joint won't limit the weight that the muscles of the glenohumeral (pecs) and elbow joints (triceps) can handle. But in the pushup, the weight is always loaded on the ribcage, meaning it isn't possible to avoid loading the AC joint.
Hence scapular retraction has no advantage in a pushup. And if that's the case, then you might as well actually deliberately protract the scapula at the top of a pushup, in order to also get a training effect on the serratus anterior.
As for your more general question about keeping the shoulder blades retracted during resistance exercise in general, there's no answer to this that is applicable to all exercises. It's common to keep the shoulders retracted and depressed in the bench press for the reasons described above, and in the squat the shoulder are kept retracted and possibly elevated because that bunches the muscles up into a pad that will reduce the discomfort of the barbell pushing against bones, and will keep the bar locked in position on the back. But the fact that shoulder retraction is beneficial in these two exercises (in different positions of elevation and for wholly different reasons) does not mean that there's any need to keep them retracted in other exercises.