B737 CREW HAD CTLR; PLT FLT NUMBER MIX-UP.

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I WAS FO AND PF. AUTOPLT WAS ENGAGED. ACFT CLBING IN RELATIVELY SHALLOW; HIGH SPD CLB PROFILE. WE WERE CLBING TO ASSIGNED FL330 WHICH WAS IN ALT WINDOW. AT APPROX FL293 ATC QUERIES 'ACFT X VERIFY LEVEL AT FL290?' BEFORE WE COULD RESPOND ANOTHER COMPANY ACFT JUMPED IN: 'THAT'S AFFIRMATIVE; FL290; ACFT Y.' ATC: 'I SHOW YOU A COUPLE HUNDRED FT HIGH.' ACFT Y; 'NEGATIVE SIR; WE'RE LEVEL HERE AT FL290.' ATC: 'AM I TALKING TO ACFT X?' MEANWHILE; DURING THIS CONFUSING INTERCHANGE; I'VE CLICKED OFF THE AUTOPLT AND INITIATED A DSCNT AS I'VE NOTICED AN OPPOSITE DIRECTION MD80 WHICH IS OBVIOUSLY A POTENTIAL THREAT AND THE REASON FOR THE CTLR'S CONCERN. WE REACH APPROX FL297 DURING THE PUSHOVER FROM CLB TO DSCNT AND RECEIVE A 'TFC' CALLOUT FROM OUR TCASII SYS (NO RA). AFTER SEVERAL RADIO BLOCKS FROM OUR COMPANY FLT # ACFT Y; THE CAPT WAS FINALLY ABLE TO RESPOND AND RPT LEVEL AT FL290; AND THAT WAS THE FIRST TIME WE HAD EVER READ BACK FL290. THAT WAS THE END OF THE INCIDENT AND THE BEGINNING OF A MISERABLE EVENING OF FLYING; WONDERING WHAT HAPPENED AND WHAT WOULD COME OF IT. IN HINDSIGHT I CAN SEE SEVERAL ERRORS. THE CAPT AND I BOTH AGREE THAT WE NEVER READ BACK ANY CLRNC TO FL290. MY SUSPICION IS THAT WHEN ATC ORIGINALLY ASKED US TO LEVEL AT FL290 RATHER THAN FL330 IT WAS REPLIED TO BY ACFT Y. THIS MISTAKE SHOULD HAVE BEEN CAUGHT BY US OR ATC; BUT IT WAS NOT. BETTER RADIO SITUATIONAL AWARENESS ON OUR PART WOULD HAVE SOLVED THE PROB. I WILL SAY; HOWEVER; THAT I DON'T RECALL COMMON ATC PRACTICE USED WHEREBY THEY ALERT CREWS TO SIMILAR SOUNDING CALL SIGNS. THIS SHOULD HAVE BEEN DONE; ESPECIALLY WITH THE SIMILARITY OF ACFT Y AND ACFT X. THIS IS A REAL PROB FOR OUR ACR. YOU MAY HAVE 6 CALL SIGNS IN AS MANY HRS AND ALL OUR FLTS ON FREQ ARE SIMILAR. I'M SURPRISED THIS DOESN'T HAPPEN MORE OFTEN.