When Prince George spends the night at his great-grandma's house, which is, of course, Buckingham Palace, Queen Elizabeth II goes out of her way to spend time with him. She also selects little presents that she leaves at the foot of George's bed when he spends the night.
Another thing that bonds young George and his great-grandmother is the fact that he's growing up with a spotlight on him. As third in line to the throne, there's great expectation on his young shoulders, something Elizabeth grew up with as well.
While George may not totally understand the idea of being a monarch yet, he "understands" that he's "different from his younger brother and sister," as a royal expert explained. "After all, he gets taken away to do photoshoots with Her Majesty the Queen, so he must notice he's been singled out."