We could rewrite it as 'Naruto owns a new ninja training house in fanfiction'. This way, it focuses on Naruto's role as a leader and teacher in the ninja world, which is more in line with his character.
There could be many possibilities. It might be an original character created specifically for the fanfiction. However, it could also be a well - known house - elf from the original series, like Dobby, who in the fanfiction somehow comes to be owned by Harry Potter in a different set of circumstances. Maybe Harry saves Dobby in a more complex way in this fanfiction and as a result, Dobby becomes his house - elf.
I don't think it's a super popular concept in the Naruto fanfiction community. There are so many directions fanfiction can take. Some focus on romance, others on battles or new powers. While Naruto's relation to Konoha is fundamental, the idea of him 'owning' it in fanfiction is not something that seems to be everywhere. That said, there may be some fanfictions out there that play with this idea, but compared to things like exploring Naruto's heritage or his growth as a ninja, it's probably not as prevalent.
I'm not sure exactly who owns the 'A Christmas Story' House. It could be a local entrepreneur or a collector of movie memorabilia. The details of the ownership might not be widely publicized.
The ownership of the Christmas Story House might vary. Sometimes, it could be a private individual or a company that specializes in preserving and managing such properties.
Maybe he owns a ninja training academy as a business. Since he is a powerful ninja, he can pass on his skills and knowledge to the next generation. He could offer different levels of training courses for both beginners and more advanced students.