Naruto calling Tsunade 'grandma' is one of those hilarious yet oddly fitting nicknames that just stuck. It started as a playful jab at her age—Tsunade is significantly older than she looks due to her chakra-enhanced youthfulness, and Naruto, being his blunt self, noticed. But there's more to it than just teasing. Tsunade was the Third Hokage's student, and Jiraiya (who was like a grandfather figure to Naruto) was her teammate. So, in a weird way, there's almost a familial connection there. Naruto's never been big on formal titles, and 'grandma' just felt more personal than 'Lady Tsunade.' Plus, it perfectly captures their dynamic: she scolds him like a grandma, and he grins back like a cheeky grandson.
What I love about this is how it reflects Naruto's character—he disrespects authority in the most endearing way possible. Even when he becomes Hokage, he keeps that irreverent charm. Tsunade might roll her eyes, but you can tell she secretly finds it amusing. It’s like when your little cousin gives you a ridiculous nickname, and eventually, even your parents start using it.