THE FLC OF A B737 FINDS THEMSELVES NOSE TO NOSE ON TO A COMPANY FLT ON THE TKOF ROLL; OPPOSITE DIRECTION; ON RWY 31C AT MDW; IL.

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WHILE HOLDING SHORT OF RWY 13C IN MDW; WE (FLT 5XY) HEARD TWR CLR FLT 3XY INTO POS AND HOLD ON RWY 13C. WE ASSUMED THE CALL SIGN TO BE IN ERROR AND THE FO READ BACK THE CLRNC WITH EMPHASIS ON THE 5XY TO CORRECT THE CALL SIGN. TWR DID NOT RESPOND AND WE HAD NO INDICATION WE WERE BLOCKED; SO WE PROCEEDED ON TO RWY 13C WITH A SLIGHT BACK TAXI TO GET FULL LENGTH. DURING THIS SOMEONE WAS CLRED FOR TKOF. THE FO AND I WERE CONFUSED BY THE CLRNC BUT NOTICED LNDG LIGHTS AT THE OTHER END OF THE RWY WHEN WE GOT ALIGNED; SO WE BEGAN TO EXIT THE RWY WHEN TWR CALLED FOR THE ACFT ON THE TXWY (US) TO EXIT. CONTRIBUTING FACTORS INCLUDE: 1) SIMILAR CALL SIGNS THAT WE WERE NOT AWARE OF. 2) SIMILAR SOUNDING RWY; IE; RWY 13C AND RWY 31C. 3) OPPOSITE DIRECTION DEP WITHOUT NOTICE TO US. 4) ASSUMING THAT WE WOULD BE NEXT FOR RELEASE AND NOT GETTING POSITIVE CONFIRMATION OF THE CLRNC. SUPPLEMENTAL INFO FROM ACN 456449: ASSUMING THAT THE XMISSION WAS MEANT FOR US; WE ANSWERED INQUISITIVELY 'ROGER; THAT'S ACR FLT 5XY; POS AND HOLD RWY 13C;' AND CAUTIOUSLY PROCEEDED ONTO RWY 13C. ACFT FLT 3XY CONTINUED TKOF ROLL AND DEPARTED NORMALLY. CONTRIBUTING FACTORS: THE USE OF RWY 31C FOR DEPS WHEN RWY 13C WAS THE ACTIVE; AND NO CORRECTION BY TWR WHEN WE REPEATED WHAT WE THOUGHT WAS INSTRUCTION TO POS AND HOLD ON RWY 13L. I WOULD ASSUME THAT IT WAS STEPPED ON AS OTHER FLT RESPONDED SIMULTANEOUSLY. A HIGHER DEG OF SITUATIONAL AWARENESS BY US AND A NO ASSUMPTIONS ATTITUDE WOULD HAVE PREVENTED THIS SIT.