My name still the same (US Passport, DNI or Birth Certificate) and I they never asked me about my DNI, they just see "Argentina" under place of birth and that's all, it was the same thing when I crossed (by land) from Argentina to Chile .
The only time they were not aware/sure of that is when I took the ferry from Montevideo to Buenos Aires (Dec 2013), the lady didn't how what to enter in the computer, then she asked to a colleague to confirm it.