As I understand he will always be Argentine in the eyes of Argentina and thus eligible to apply for a passport. Argentine citizenship by birth can never actually be renounced, so he already has it with or without the passport.
Have a closer look into the country where you aim on settling down as few countries that allow multiple nationalities in the first place actually draw the line at having only two nationalities/ citizenships.
"Dual citizenship" usually only applies as a concept to countries where the rule does not generally permit holding other nationalities unless you fall into a loophole (e.g. children of Austrians, Germans, Koreans, etc born in Argentina where citizenship cannot be renounced.)
Some countries that permit multiple nationalities simply say after you have naturalized as a citizen you are not allowed to acquire further citizenship (like Brazil, Ireland etc.) but it does not usually apply if citizenship is based on decent. On the other hand, some countries say if you want to naturalize as a citizen then you must renounce previous citizenship if possible but they still allow natural-born citizens to acquire other nationalities.
These days multiple nationalities is commonplace - having three or four passports is not that hard for many people and rarely raises an eyebrow. It's generation globalization.
PS - There is also an advantage to Argentine-born Argentine citizens wishing to naturalize in Spain (and thus fast-track EU citizenship)
This has been a really helpful reply! Spain is actually the country we wish to settle in. Could be a 13 years or so left in this career (hopefully) but we do want to live in Spain! What is the advantage could you explain more? Thank you so much.