ATC USES WRONG CALL SIGN WHEN VECTORING; CAUSES TCASII RA.

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WE WERE ON A 190 DEG HDG AT 4000 FT 8-10 MI OUT OF IAD BEING VECTORED FOR A RWY 1L ILS APCH AT IAD WASHINGTON BY AN IAD CTLR IN IMC CONDITIONS. WE WERE GIVEN A TURN TO 30 DEGS TO INTERCEPT THE 1L LOC AND DSND TO 3000 FT. IT WAS A LATE TURN. OUR FLT NUMBER WAS ABC AND THE CTLR KEPT GETTING US MIXED UP WITH SAME ACR ACC. WE PASSED THROUGH THE LOC BEFORE WE HAD EVEN TURNED 90 DEGS. WE WERE AT OUR COMPANY'S MAX BANK ANGLE TO INTERCEPT. WE SAW ANOTHER ACFT ON THE TCASII TURNING INTO US. HE WAS AT OUR 11 O'CLOCK POS. WE GOT A TA ALERT BUT DUE TO IMC CONDITIONS WE DIDN'T SEE HIM. AT 3300 FT ALT; WE GOT AN RA TO CLB AT 2000 FPM AND WE COMPLIED. WHEN WE GOT THE CLR OF CONFLICT ON THE TCASII; WE RPTED THE DEV TO ATC. THEY CALLED THE WRONG FLT NUMBER AND ACR ACC TOLD THEM IT WAS US; ACR ABC. WHEN WE GOT THE CTLR STRAIGHT ON THE NUMBERS; WE WERE VECTORED FOR ANOTHER APCH. I FEEL THERE WAS NOT ENOUGH TFC SEPARATION AND THE CTLR SHOULD HAVE RECOGNIZED THE DANGEROUS SITUATION DEVELOPING AND KEPT US APART FROM EACH OTHER'S FLT PATH.