Setting aside the public ownership of land (I don’t think we would differ as much as you think).

No there is no difference. The issue here is you focusing on the volcano part. It’s a land mass like any other. The article didn’t go into history of ownership but it’s possible that it was passed down and likely from before the USA was a thing and well before Hawaii was part of the USA.

I can guess without looking that if I did a search for privately owned mountains I would find more than a few. Same goes for the aforementioned private islands that can be bought by anyone with enough cash/credit.

What about people owning lakes? If I took an acre of my land (yes, we are buying 3+acres) and turned it into a lake, would that seem odd to you? Would you want me to open my land up to anyone who wanted to visit, fish, etc? Or would that not be as absurd?

Do yourself a favor and ignore the volcano part, it will make much more sense.