Ramp Crew Chief reported communications issues and failure of ATC to coordinate simultaneous push backs on adjacent gates led to a sudden stop to prevent a collision between aircraft. The Ramp Crew Chiefs pulled aircraft back and re started the push backs with closer coordination and the aircraft departed.

Narrative

I was the Crew Chief on Gate XX working Aircraft X to ZZZ. After an uneventful upload my Wing Walkers (Person A and Person B) were in place and I contacted the Flight Deck for push clearance. The Pilot received clearance and relayed the instruction 'Clear to push; pull to Spot XX.' Aircraft X was on an A320 but after countless pushes of 321s out of Gate XX; when I commenced my push pull I pointed the tail of the aircraft at Gate XY (pushing a A320 tail south out of Gate XX is possible but a tight push that can end with the APU ft. away from carts at T Point). Gate XZ was given clearance to push Aircraft Y at the same time. At this point my A320 and Person C's 737 were on a collision course. Thankfully the Wing Walkers all did their job and Gate XZ halted their push while I pulled my aircraft out of the way.During my push from Gate XX and Person C's push from Gate XZ we were both in each other's blind spots. The nose of each aircraft blocked the view of the other. Once the nose of my aircraft cleared I saw the 737 and I realized what was happening. Person C was on top of it and immediately stopped his push while I cut my push tighter to clear.When Gate XX was retrofitted for A321s [Maintenance Crew members] were told by Tower and Management that Gate XZ would not be allowed to push while Gate XX was commencing a push/pull. This was because Gate XX would have to point the tail of the 321 at Gate XY (or at least wide of line XY) to avoid T-Point. A tail south push of A320s is possible but not ideal as it is very tight quarters.After the near miss I called the Tower and asked to speak to Ramp Control. That is when Person D informed me that it was SOP for Gate XX and Gate XZ to push simultaneously. She said I should have pushed my aircraft completely tail south before towing to Spot XX. I asked her 'If I was pushing a 321 would you have given Gate XZ clearance to push?' The answer was no. I believe it is dangerous to have different push expectations out of the same gate with different aircraft while the push back verbiage remains the exact same.Do not allow Gate XZ and Gate XX to push at the same time.