CHT LTT WRONG ARPT APCH AT SEA. SHOULD HAVE BEEN ON APCH TO BFI.

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I COPIED AND READ BACK THE IFR CLRNC OVER THE ACFT RADIO WITH VANCOUVER CLRNC WITH THE EXCEPTION THAT OUR DEST WAS INCORRECTLY GIVEN AS SEATTLE-TACOMA ARPT (KSEA) RATHER THAN BOEING FIELD. CLRNC STATED HE WAS BUSY AND WE COULD TKOF ON THE EXISTING CLRNC AND THEN MAKE THE DEST ARPT CHANGE WITH SEATTLE CENTER WHEN WE TALKED TO THEM. WE ELECTED THE LATTER COURSE. WE WERE RECLRED BY SEATTLE CENTER TO BOEING FIELD VIA THE JAWBN 4 ARR RTE. ENRTE; I RECEIVED BOEING ATIS INFO XYZ. THE LOC BACK COURSE RWY 31L WAS IN USE. ON INITIAL CONTACT WITH APCH CTL AT SEATTLE; I RPTED RECEIVING INFO XYZ AT BOEING SAYING 'XYZ' AFTER OUR N NUMBER AND ALT. WE WERE GIVEN A LOWER ALT AS A XING RESTRICTION AND THEN HANDED OFF TO ANOTHER CTLR. I EXPECTED TO BE GIVEN A RADAR VECTOR FOR A VISUAL APCH RATHER THAN CONTINUE ON THE STAR FOR A BACK COURSE APCH; BUT NONE WAS GIVEN BY APCH. COMPLETING THE IN RANGE CHKLIST; I CHKED THE JAWBN ARR TO SEE WHAT WE SHOULD DO FOR A RTE AT ALKIA INTXN. IT DEPICTED A TRACK CONTINUING STRAIGHT AHEAD TO SEA AND A RADAR VECTOR HDG OF 160 DEG. I TOLD THE FO FLYING TO CONTINUE STRAIGHT AHEAD UNTIL I VERIFIED WITH THE CTLR IF HE WANTED US ON A VECTOR HDG. HE CAME BACK ASKING IF WE SAW ALKIA INTXN. I RESPONDED THAT WE DID AND AS WE WERE NOW SEVERAL MI INSIDE IT I ASKED WHAT HDG HE WANTED US ON. HE CAME BACK SAYING HE NEEDED THAT TURN TO 160 DEG AT ALKIA FOR SEQUENCING; OBVIOUSLY UPSET; AND GAVE US AN ALT AND AIRSPD AND HDG. WE WERE NOW ON VECTORS FOR AN APCH WITHOUT BEING TOLD WHAT APCH; BUT I KNEW IT EITHER HAD TO BE A BACK COURSE OR VISUAL; EITHER ONE OF WHICH WE WERE PREPARED FOR. WE WERE GIVEN ANOTHER FREQ CHANGE; AIRSPD CHANGE AND ALT CHANGE AND ASKED IF WE STILL HAD 'THE ARPT IN SIGHT.' I VERIFIED THAT THE FO STILL HAD THE ARPT AS IT WAS ON HIS SIDE OF THE AIRPLANE. I RPTED 'ARPT IN SIGHT' AND WE WERE CLRED FOR A VISUAL APCH. I READ BACK 'CLRED FOR THE VIS APCH.' I WAS RESPONDING TO CALLS FROM THE FO FLYING AND RUNNING THE LNDG CHKLIST AND LOOKED OUT THE WINDSHIELD TO SEE US APPARENTLY OVERSHOOTING THE TURN TO FINAL APCH. I QUERIED THE PF WHO RESPONDED THAT WAS SEA-TAC. I LOOKED AT THE RWY CONFIGN AND VERIFIED HE WAS CORRECT AND THEN THE CTLR CAME ON ASKING WHAT ARPT WE WERE LNDG AT. I RESPONDED WE WANTED TO LAND AT BOEING. HE CAME BACK WITH A CLRNC FOR A VISUAL AT BOEING AND TOLD US WE NEEDED TO CORRECT HIS CLRNC TO THE WRONG ARPT. I HAD TIME ON THE FREQ ONLY TO SAY THAT WE WERE BUSY AND WERE GIVEN A FREQ CHANGE TO BOEING TWR AND LANDED. I DID NOT CATCH THE CTLRS MISTAKE. HE DID NOT SAY WHAT ARPT HE WAS CLRING US TO. IF HE SAID THE RWY NUMBERS; I DID NOT CATCH HIM SAY 34L RATHER THAN 31L; AS IT SHOULD HAVE BEEN AT BOEING. IT IS MY PRACTICE TO READ BACK CLRNCS AS RECEIVED TO VERIFY RWYS L AND R; BUT I READ BACK ONLY 'CLRED FOR THE VISUAL' IN THIS INSTANCE SO EITHER THE CTLR DID NOT SPECIFY OR I DID NOT CATCH IT. SEATTLE CENTER DID NOT PASS ON THE NEW CLRNC TO SEATTLE APCH FOR A BOEING LNDG.