Air carrier Captain reported confusion by ATC believing Captain was being referenced for taxi instruction resulted in confusion during taxi.

Narrative

We were at spot XX waiting to call ground for taxi. Ground made a call; 'Spot XY?' before we were able to get a call in. There was nobody at spot XY at the time; and we were the only one in the vicinity; so we responded with our call sign thinking he was referring to us. Ground told us to give way to two jets and then taxi out. We waited for the two aircraft to pass and then began our taxi. ATC told another aircraft to give way to us. They complied and allowed us to taxi out. As we were pulling out of spot XX; ATC told spot XY to turn left and begin their taxi. As I was making the turn onto taxiway K; another ATC voice (maybe an instructor) came on and said 'Are you gonna give way?' with no call sign. ATC then came on and said 'Airline XXXX you were supposed to give way but you cut them off.' That crew was confused and said something along of the lines of 'We are giving way.' ATC seemed to think otherwise. The aircraft at spot XY mentioned our call sign and ground said 'I think you're right.' My First Officer (FO) told ground that we never heard an instruction to give way; to which they did not respond. We continued the rest of our taxi with no issue. A couple of minutes passed and the aircraft that ATC thought had made the mistake queried ground about the situation and ground responded with 'Don't worry about it;' leading me to believe that they had gotten call signs mixed up. There were multiple of the same airline flights in the vicinity and ATC was confused as to who was who. Not clarifying who was at which spot added to the confusion. When ground called up asking who was at spot XY; we could have clarified that we were at spot XX to avoid any confusion. We did not know that there was an aircraft behind us making their way to XY shortly after. Ground could have been more clear with their instructions and included the correct call signs with each instruction. With spots XY and XX in very close proximity and one aircraft (us) having already received taxi instructions; I think it makes more sense to let the aircraft that's already moving get out first.