I've only just seen it, but judging by the "own" I'm guessing its a way of strongly associating the sandwich as being the girl's. If Suoi ("his" or "her") was used, it would be ambiguous whether it was the girl eating her own sandwich or someone else's of the same or maybe the opposite sex. Propri says that its definitely hers. Hope this helps!
It means 'own' as in 'her own', to emphasize that she could have eaten someone else's sandwich, but she chose her own. Or that she usually steal other people's sandwiches, but today she made an exception and ate her own. Never seen it mean 'really', so I can't comment on that.
One of the translations I got (in the sentence, not the hints) for "panini" is ... panini! In the US, a panini refers to a very specific type of sandwich wherein some sort of filling is put inside some sort of bread then pressed or grilled or otherwise heated so that the exterior is crunchy. The impression I got is that the Italian word refers to the generic sandwich, not this very specific type of sandwich.
Can anyone clarify?
here is a site that can help but each of you has to translate it on his own language because i am greek and i do not want to make any mistakes by explaining it!
Italian possessive adjectives agree in gender (masculine/feminine) and in number (singular/plural) with the noun they refer to. For instance, "horse" is a masculine noun, (regardless of the gender of the actual horse in question) so it uses the masculine mio and miei.