I don't know that I would consider this an answer so much as a reflection and suggestion about how to approach this idea in Japanese. (If the moderators feel this is unsuitable as an answer, then please let me know, I will remove it.)
As I mentioned in a comment to your post, "to be someone's person" is a highly idiomatic phrase made popular to some extent by--perhaps even coined in--Shonda Rhimes' Grey's Anatomy.
"My person" is the person I can rely upon in a crunch or a crisis; it's the person who will be there no matter what happens, no questions asked, no judgment. It's certainly not synonymous with "best friend" or even "BFF"; neither really do "soul mate" and "kindred spirit" capture this feel either. I mention these other notions of friendship since they have been around a while and can probably be reasonably translated into Japanese (though off the top of my head right now, I don't how best to translate these concepts either).
In my comment, I mused about capturing this notion through the idea of a bodhisattva, 菩薩{ぼさつ}. But, almost from the moment I made that comment, I felt it missed the mark. It seems to me that in English, in circles where folks understand the concept of a bodhisattva, we use the word much more casually than Japanese perhaps do 菩薩. (I certainly welcome feedback on this point.) 菩薩 somehow, to me at least, has a more technical or stiff or formal feel about it; whereas in English, bodhisattva has a much easier feel to it. But even with that said, bodhisattva doesn't exactly capture the feel of my person either.
I don't think you're going to find a good single expression in Japanese to capture this. I would recommend that you identify for yourself the quality of your person which best captures this feeling and express that instead.