TOKYO — Wherever the Japanese animated filmmaker Makoto Shinkai goes these days, he gets some variation on the question of what it’s like to be the Next Big Thing.

Specifically, Mr. Shinkai, whose “Your Name” is the top-grossing movie of the year in Japan, is frequently asked how he feels about being the successor to Hayao Miyazaki, who has towered over Japanese animation for nearly three decades as the beloved creator of critically acclaimed hits like “Princess Mononoke” and the Oscar-winning “Spirited Away.”

“I feel very honored, but I think it’s an overestimation,” Mr. Shinkai, 43, said during an interview at the offices here of Toho, the Japanese film production and distribution company, graciously rebuffing the coronation. “I don’t think anybody can replace Mr. Miyazaki.”