Commercial Pushback Operator reported that they received non standard push instructions from Captain that lead to a ground conflict during pushback.

Narrative

The Captain from Aircraft X [at] ZZZ on [gate] X had given me the clearance to pushback the aircraft. The Captain stated: 'Brakes released; they (the Tower) said 'not too deep''. I responded: 'Acknowledged; commencing the push.' I started the push keeping an eye out for any aircraft in the alley. I had noticed that an aircraft was indeed pushing back from around B4/B6; so I was being mindful not to get to close when I complete my turn off of X. Halfway off of the gate; I noticed that [gate] Y was pushing back as well and had already begun their turn into the alley. I started to slow down; but the necessary force to stop the aircraft would have jolted everyone on board. I alerted the flight crew of the situation by saying: ' It looks like the Tower also cleared Y to push as well; I am going to have to maneuver you to a different location.' The Captain responded: 'So that's what they meant by not too deep! Well do what you can but don't make it sound expensive'. I responded: 'I'll try.' Using the momentum I still had; I started to turn the aircraft's tail south away from the alley. The only safe place I could put the aircraft was at the drop point; usually intended for use by [gate] Z. After we approached the drop point; the rest of the pushback procedure went by the book. When I returned to the gate; I spoke to the crew chief about what had transpired. The crew chief was watching from the beginning; but just wanted to know what instruction the flight crew had given me. During or next flight; the manager had approached the Crew Chief about what happened; stating that the Tower had expressed concern about the pushback.While halfway off the gate; I realized the second aircraft pushing back from Y. My attention was at maintaining my distance from the first aircraft that was further down the alley. Cause: The event occurred because the Tower's instruction of 'not too deep' was vague; leaving me to my own observations to deduce the Tower's meaning.Response: After the pushback was finished; I had alerted my Crew Chief of the situation.Suggestions: Instead of a vague order; the Tower should have conveyed a directive more clearly. Had I been aware that [gate] Y was going to push at the same time I would have started my turn sooner to accommodate a simultaneous push into the alley. As I stated earlier; 'not too deep' was such a vague instruction that even the flight crew sounded surprised when I alerted them of the near collision.