Neither Cirdan nor Celeborn was eligible to be King of the Noldor, seeing as neither of them were Noldorin. Cirdan was a Teleri, and Celeborn was a Silvan elf, of Thingol's brethren.
As for why Elrond didn't lay claim to any sort of Kingly title, I think it was because the so-called Last Alliance of Elves and Men was indeed the last time the elves would use strength of arms to thwart evil. Since the remnant of the Noldor was but a shadow of what it had once been, no further wars were possible, and no King was needed.
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