It's flat out wrong.
We had no issue in either of our cases with the clerks, secretaries, judges. I'm also being recognized as Italian since birth (jus sanguinis). My wife is going to BA in the next 6-8 weeks to do option of Dutch recognition (effective the date she takes oath, not since birth. Her mom was from NL and my wife is considered 'latent Dutch'). In that case, the order is important. Had she opted for Dutch first and then became Argentine--she would have lost her Dutch citizenship (unless perhaps I was already Argentine). Where Argentina does not allow renunciation so she can keep it. In theory, she can also get Italian by marriage. NL is quite strict, but lets spouses obtain other citizenship(s) of spouse as well.
Granted, that's a bit ridiculous to have four (but possible in some cases). My youngest is US/IT/AR since birth and NL 'latent Dutch' The other one is US/IT and NL 'latent Dutch' and we're working on her Argentine option which must be completed before we opt for Dutch recognition. We're on the fence as to whether we do both EU citizenships.
It was through doing the Argentine process that I learned a good deal about citizenship law and only at the very end of the process--revisited our respective European options. It was not our intent to collect passports like others collect passport stamps. However, we like the idea that our girls will have rights to work, study, live on three continents (under current treaties, provided EU and Mercosur hold together).