G4 IN POS ON RWY 13L AT DAL INITIATES A SIMULTANEOUS TKOF WHEN TFC ON RWY 13R WITH SIMILAR CALL SIGN; IS ISSUED DEP CLRNC.

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GULFSTREAM 4 WAS CLRED INTO POS AND HOLD AT DAL RWY 13L. WE ACKNOWLEDGED THAT CLRNC AND WERE IN POS FOR APPROX 30 SECONDS WHEN WE HEARD OUR CALL SIGN WITH A TKOF CLRNC TO EXPEDITE; WHICH WAS ACKNOWLEDGED AND COMPLETED WITHOUT INCIDENT AND OR FURTHER INSTRUCTIONS FROM THE TWR. ONCE AIRBORNE WE HEARD THE TWR CONTACT A LEARJET WITH A SIMILAR SOUNDING NUMBER TO ADVISE HIM OF A G4 OFF RWY 13L. THE LEAR TOOK OFF FROM RWY 13R. HE ACKNOWLEDGED VISUAL CONTACT. TWR POINTED THE LEARJET OUT TO US AND WE ALSO HAD A VISUAL ON HIM. AFTER FLYING A PARALLEL COURSE WITH THE LEAR FOR SEVERAL MI; DAL TWR HANDED US OFF TO DEP CTL. SHORTLY AFTER ESTABLISHING CONTACT WITH DEP WE WERE GIVEN A 15 DEG TURN TO THE L. AFTER FLYING AT CRUISE ALT FOR ABOUT 45 MINS WE WERE ADVISED BY CTR TO CONTACT DAL TWR; BY PHONE; WHICH WE DID. AT THAT TIME WE WERE INFORMED THAT WE HAD MISUNDERSTOOD OUR TKOF CLRNC. THE TWR ACKNOWLEDGED THE SIMILARITY OF THE NUMBERS. G4 XXXXX; LEAR NYXXXX. NOTWITHSTANDING THE APPARENT MISUNDERSTANDING; AT NO TIME WAS THERE EVER A SAFETY ISSUE BECAUSE BOTH CREWS ACKNOWLEDGED VISUAL CONTACT FROM TKOF TO THE TIME WE WERE HANDED OVER TO DEP CTL.