I don't know that the concept is unique to Japanese culture or Anime/Manga. There's a couple of concepts this touches on
The first would be, what TV tropes calls, Cannibalism Superpower. The name is a little deceptive, in that real cannibalism involves eating the flesh of your fellow beings of the same type (which generally means you've killed them first). The broader concept here is that the character with this power absorbs the power of the other beings they consume or somehow merge with. So Howl eating Calcifer, or Ponyo eating human blood to become human fits here. The TV Tropes site lists at least a dozen instances of this in Anime alone, but there's a lot of others as well.
The second is this: food has a general significance in everyone's lives, and it's something most societies do communally ever so often (i.e. Thanksgiving). If you've never seen it (and it's a somewhat obscure movie), you need to watch Babbette's Feast. The general gist of the movie is there's a woman (named Babbette, obviously) who wins the lottery and wants to cook a feast for the two sisters she works for. To say that's an oversimplification would be an understatement. The movie turns a meal into something far deeper. It connects the characters in a unique and fundamental way with food (I've never seen another movie like it, although Eat, Drink, Man, Woman came the closest).
When I watch Miyazaki, I think he's simply building on this simple truth. I'm not claiming that he's been influenced by that film, but he's trying to make the characters relate-able. Another movie that did this was Summer Wars where one main character is staying in the home of the other. They share meals (amongst other experiences) in the movie and it's a way to show their bonds and draw you, the audience member, inside.
When the grandmother dies, you, like the main character who is the guest, will likely feel a sense of loss even though she is not your relative.
The meals help with that aspect of the story. Miyazaki is masterful in making you care about his characters (which is why his movies are so popular).