A POTENTIAL RWY INCURSION AND GAR DURING LOW VISIBILITY AND SNOW CONDITIONS.

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MY CREW ARRIVED AT GATE X AT XA00 FOR AN XA50 DEP TO HPN. DUE TO SNOW AND POOR VISIBILITY; OUR INBOUND FLT DID NOT ARRIVE AT THE GATE UNTIL APPROX XA30. AFTER THE INBOUND CREW AND PAX WERE UNLOADED; WE GAINED ACCESS TO THE ACFT AND BEGAN OUR PREFLT DUTIES. THE ACFT WAS BEING CATERED AND FUELED AS PAX STARTED TO BOARD. THE FLT DECK WAS SET UP FOR DEP AND A CALL MADE TO REQUEST A DEICE AS THE TEMP WAS -2 DEGS C WITH SNOWFALL. THE DEICING PERSONNEL ACKNOWLEDGED OUR REQUEST AND PROVIDED US WITH 2 SEPARATE FREQS; ONE FOR THE RWY 30L DEICE PAD; AND THE OTHER FOR THE RWY 12L DEICE PAD. AT APPROX XA55; THE REFUELING WAS COMPLETED. WE CONTACTED MSP GND CTL FOR A PUSHBACK CLRNC. WE RELEASED THE BRAKE AT XA57 THEN WERE PUSHED BACK. ALL 4 ENGS WERE STARTED AND WE REQUESTED A TAXI CLRNC. WE WERE INSTRUCTED TO TAXI TO THE RWY 12L DEICE PAD. AS WE WERE TAXIING OUT; WE NOTED THERE WAS SO MUCH CONGESTION ON THE GND FREQ THAT THE GND CTLR WAS ISSUING CLRNCS TO ACFT AND STATING 'HOLD THE READBACK.' AFTER THE DEICING WAS COMPLETED; WE CONTACTED GND AND WERE ISSUED A CLRNC TO TAXI TO RWY 30R VIA TXWY P; HOLD SHORT OF TXWY M. WE READ BACK AND COMPLIED WITH THE CLRNC. UPON ARR AT TXWY M; WE WERE GIVEN A FURTHER CLRNC TO TAXI TO RWY 30R VIA TXWY P. WE RESPONDED TO GND; THEN CONTINUED OUR TAXI. AS WE APCHED THE END OF TXWY P; GND ASKED IF WE WERE PAST TXWY P3. WE RESPONDED THAT WE HAD PASSED TXWY P3. WE SWITCHED OVER TO TWR FREQ AS WE APCHED THE END OF TXWY P. THERE WAS 1 ACFT IN FRONT OF US AS WE WAITED FOR OUR DEP CLRNC FOR RWY 30R. THE ACFT IN FRONT OF US WAS ISSUED A DEP CLRNC. THEY TAXIED INTO POS; THEN DEPARTED RWY 30R. WE THEN HEARD MSP TWR CLR US INTO POS AND HOLD. THE FO READ BACK THE CLRNC. TWR RESPONDED BACK TO US 'ROGER.' AFTER BEING IN POS APPROX 1 MIN; WE HEARD TWR ISSUE A CLRNC TO THE EFFECT 'ACFT Y ??? TURN L HDG 2-6-0 CLRED FOR TKOF.' BECAUSE THE CTLR HAD NOT SPECIFIED RWY 30L OR RWY 30R IN THE CLRNC TO ACFT Y; WE ASSUMED ACFT Y WAS ON RWY 30L (WHICH HAS A SEPARATE FREQ AND THAT TWR WAS HANDLING BOTH FREQS). APPROX 15 SECONDS LATER; TWR ISSUED ANOTHER TKOF CLRNC TO ACFT Y. I THEN STATED TO THE FO WORDS TO THE EFFECT 'TWR MUST THINK WE ARE ACFT Y.' WE IMMEDIATELY TRIED TO ADVISE HER THAT ACFT X WAS IN POS AND HOLDING ON RWY 30R. I BELIEVE WE (ACFT X AND TWR) STEPPED ON EACH OTHER. WE AGAIN CALLED THE TWR AND ADVISED 'ACFT X IS IN POS AND HOLDING ON RWY 30R.' SHE RESPONDED 'ACFT X; I DID NOT CLR YOU INTO POS AND HOLD ON RWY 30R.' SHE THEN CONTACTED ANOTHER ACR ACFT THAT WAS ON THE APCH AND TOLD THEM THERE WAS TFC ON THE RWY AND TO GO AROUND. ABOUT 1 MIN LATER; WE RECEIVED A CLRNC 'ACFT X; FLY RWY HDG; CLRED FOR TKOF RWY 30R.' WE READ BACK THE CLRNC THEN STARTED OUR TKOF ROLL. AS WE WERE ROLLING; WE HEARD TWR CLR ACFT Y INTO POS AND HOLD RWY 30R. THIS WAS NOT CORRECT AS THE ACFT IMMEDIATELY BEHIND US WAS AN ACR AIRBUS. I CAN SEE NUMEROUS FACTORS THAT CONTRIBUTED TO THIS SIT. 1) THE FLC WAS TRYING TO GET THE FLT OUT OF THE GATE ON TIME. THIS WAS A STRESS FACTOR CAUSING THE CREW TO BE ANXIOUS. 2) THE GND CTLR ADVISING ACFT TO 'HOLD THE READBACK' MAY HAVE PREVENTED ACFT FROM ADVISING GND OF THEIR POS. 3) VISIBILITY WAS 3/4 MI DUE TO SNOW. THIS PREVENTS THE TWR FROM ACTUALLY SEEING THE ACFT THEY HAVE CLRED FOR TAXI TO OR TAKE OFF ON RWY 30R. 4) BECAUSE OUR FLT WAS FIRST IN LINE FOR TKOF ON RWY 30R; WE ASSUMED/ANTICIPATED WE WERE BEING CLRED FOR TKOF WHEN IN REALITY THE TWR CLRED ACR Y ACFT INTO POS AND HOLD ON RWY 30R. 5) TWR CLRED ACR Y FOR A TKOF WITHOUT SPECIFYING WHICH RWY. HAD TWR STATED 'ACR Y; TURN L HDG 2-6-0; YOU ARE CLRED FOR TKOF (RWY 30R OR RWY 30L)' IT WOULD HAVE BEEN APPARENT TO ACFT X AS WELL AS ACFT Y THAT THINGS WERE NOT IN ORDER.