SFO Controller described confused similar same company call sign event when the wrong strip was utilized and the wrong callsign used and read backs missed during all tower operations.

Narrative

Air Carrier XXX was #1 for departure off Runway 1R; the strip I was provided was Air Carrier XYY. The Air Carrier XYY departure strip was at Ground Control. During a extremely busy time; I taxied Air Carrier XYY into position and hold. Once traffic permitted; I cleared Air Carrier XYY to depart Runway 1R; and at about 1/2 mile upwind; I advised Air Carrier XYY to contact NCT Departure. After switching to NCT; the aircraft tagged up as Air Carrier XXX and NCT called to ask us to scan Air Carrier XXX again. The RADAR Coordinator did some coordination with NCT and we discovered that the aircraft I called Air Carrier XYY on every transmission was actually Air Carrier XXX. I have not listened to any recordings; but if Air Carrier XXX used his correct call sign I did not catch the incorrect identification. In addition Air Carrier XXX took every one of my clearances for Air Carrier XYY. Note: no loss of separation occurred as Air Carrier XYY and Air Carrier XXX were fortunately on identical departure procedures. Pay closer attention to aircraft callsign during readbacks.