I'm not sure if this is different depending on where you come from and where you emigrated to, but as a EU citizen this just doesn't click with me, at all.

We can freely move, work, marry, create companies, and everything else here already, I don't even think I'd think about it very often if I had a different passport (and it's not a thing I've only thought about for the last five minutes). (Of course this isn't true for all of them, but some of the neigboring countries).