APCH CTLR USED THE WRONG CALL SIGN WHEN GIVING A RADAR VECTOR AND WHEN THE ACFT RESPONDED THE CTLR NOTICED THE ERROR AND CORRECTED THE HEADING ASSIGNMENT.

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WE WERE ON THE SEVENTH LEG OF THE DAY--7 HRS OF FLYING AND 7 APCHS IN LOW IFR WX. THE APCH WAS THE ILS 1 AT MCI. THE ILS 1 WAS THE ONLY APCH IN USE AND IT WAS THE RUSH HOUR; SO APCH HAD A LOT OF ACFT OT DEAL WITH. WE HAD FLOWN A 25 MI DOWNWIND FOR THE APCH AND WERE INBND ABOUT 3 MI FROM THE OM WHEN ATC GAVE US AN UNUSUAL CLRNC: A LEFT TURN TO 090 DEG HDG. THE F/O READ BACK THE CLRNC; BUT WITH A RIGHT TURN; SINCE THAT WOULD HAVE MADE MORE SENSE. ATC CAME BACK AND SAID; 'NO; A RIGHT TURN TO 090;' SO I STARTED THE TURN. IT SEEMED ODD; BUT WE HAD BEEN ASKED TO SLOW FOR TFC AHEAD AND I THOUGHT WE HAD JUST GOTTEN TOO CLOSE. AS WE WERE PASSING THROUGH HDG 210; ATC ASKED US OUR HDG AND WHAT WE WERE DOING; AT WHICH POINT THE F/O DIDN'T KNOW WHAT TO SAY; SO I TOLD HIM WHAT CLRNC HE GAVE US AND THAT WE HAD READ IT BACK (TWICE). HE APOLOGIZED AND SAID HE GAVE US ABFX; A CLRNC WHICH HAD BEEN MEANT FOR ABC X. WE WERE THEN VECTORED FOR ANOTHER APCH W/O FURTHER INCIDENT. THIS EVENT OCCURRED AT THE END OF A LONG DAY OF SNOW; WITH LOW CEILINGS AND VIS; AND I EXPECT THAT BOTH THE CTLR AND I WERE SUFFERING FROM FATIGUE. HE MADE THE INITIAL MISTAKE; AND I FAILED TO RECOGNIZE IT SOON ENOUGH TO QUESTION HIM PROPERLY BEFORE THE EVENT OCCURRED. IT COULD HAVE TURNED OUT MUCH WORSE THAN IT DID. I DON'T KNOW HOW CLOSE WE CAME TO ANY OF THE MANY OTHERS IN THE AREA AT THE TIME.