Continuity mistake: Tony makes a toast at the dinner table in front of Carmella and Anthony. Watch Tony's wine glass during the toast, the amount of wine in the glass changes dramatically from shot to shot. (00:21:45)
Continuity mistake: Valentina is sitting and reading a magazine when Tony enters her hospital room. She places the magazine on her lap. Throughout the scene the magazine appears on her lap when the camera is on Tony and disappears when the camera is on Valentina. (00:18:57)
Answer: It's a mafia thing. Taking food off another made man's plate in public view like that is comparable to the guy in Casino taking off his boots and putting his smelly feet up on the table.
It is not a "mafia" thing, it's basic manners and plain gross for another person to stick their fingers into a plate of your own food. Same with the feet on the gaming table scene in Casino. It's not a "mafia" thing, it's just rude and not something a person does in public.