Natsume Youjincho (The Book of Friends) was my first anime hug, and had a lot to do with me becoming an anime fan as a result. Fruits Basket is another one. Polar Bear Cafe, also does it for me. Poyo Poyo.
I can see that March Comes In Like A Lion has a very good chance of joining this elite group as well.
Anime hugs are usually slice of life, or slice of life plus some magic. They are warm and fuzzy. Even if bad things happen, everything turns out okay. Friends are loyal and loving. Family, even adopted family, stays together and supportive. Life isn't perfect, but problems mostly are either amusing, or can be solved in an episode or two. Everybody eats well!
Now that I look at the list and think about it - they almost all have a cat! Polar Bear Cafe doesn't, but then, they have pandas. Having a cat doesn't guarantee you a spot, but it must count for something!
What is your anime hug?