The main consequence of always either 1) pushing it together OR 2) pulling it apart would be having one’s pressing become shoulder/pec dominant or triceps dominant. Other factors such as grip style will be very relevant for this development, though, and I would use the pushing/pulling strategies in conjunction with a grip style.
I would consider what exercise one is doing and why and behave accordingly to maximize the desired benefit:
For instance, if one were trying to hit max reps on flat bench and they used a closer than traditional grip, then when their muscles become fatigued it might be beneficial to imagine pushing the bar together to increase pec recruitment because a closer grip typically relies more on triceps strength.
Alternatively, if one were doing close-grip bench press for triceps development, they would probably not do the pushing the bar together strategy because it could lessen the recruitment of the tri’s.