The term, "hermit kingdom" is used to describe a country that directly shuns foreign involvement. It can describe many different countries. However, it has come to describe Korea on account of a book by William E. Griffis' book, entitled Corea: The Hermit Nation, which was published in 1882. The book was unfair as Griffis had never visited Korea or even had any experience with Korea. Moreover, there was an agenda, namely to prove that the Japanese were superior to Koreans.
In 1863 the emperor of Japan issued a decree expelling all foreigners from Japan, which resulted in minor conflicts. But this movement did not gain much traction.
In light of these two points, two conclusions can be made.
First, it is unfair to call Korea a hermit kingdom, since this term applies to many countries much more than Korea, even if it is true that Korea had little involvement with the rest of the world.
Second, since the edit which sought the expulsion of foreigners did not gain traction in Japan, Japan did not become as insular as people think. As to the question of comparison with Korea, it is close. The jury is out on this one. For example, Korea embraced Christianity, which was foreign. Japan did not. This has made a very lasting difference. In light of this, I would say that in some ways Japan was more of a "hermit" nation than Korea.