Not necessarily, you can use a build that has high crit chance to proc lots of stacks of Reen’s Ire & then do massive ignite damage (which is what I was doing here: Llama’s Firebrand/Flame Reave Spellblade Tankyboi)
That’s not true though, any melee ignite build will have Eye of Reen as it’s BIS weapon. Where you draw the line between “this weapon is BIS” & “I’m building around this weapon” I’m not sure, I think that is a matter of personal definition.
IMO, if you’re going to use a definition of “this build only works with XXX unique” to define a unique, then that could probably only be applied to a very small minority of uniques - Exsanguinous/Last Step of the Living/Shroud of Obscurity for low life builds & Dark Shroud of Cinders/Blade of the Forgotten Knight before they nerfed Fire Auras.
IMO, there are a number of uniques that, while they aren’t essential for a build to work, they really do make it significantly better - Eye of Reen for any melee ignite, Blade of the Forgotten Knight for any void hybrid/spell build, Taste of Blood/Undisputed for any fast melee bleed build, Plague Bearer’s Staff, Dragonflame Edict (probably), Draalsting, Vipertail, to name just a few.