Personal relation is one of the most natural and strongest ways to connect with something. I think for a lot of people it's intuitive and automatic and a form of empathy. To see yourself in something else, and to connect with that person or thing on a personal level, can be a moving and meaningful experience.
I don't often "self-insert" where I imagine the main character is literally me. But I like media where I can identify traits, values, and experiences that feel comparable or relatable to my own. Even if that point of comparison can see really out there. I know it when I feel it, you know?
I have a lot of feelings playing God of War 2018, for example, because of my own relationship with my father. None of the things that happen in that story literally represent anything I've gone through, but the emotional under-current is present, and I project in to that relationship. I also really love the movie Brick, by Rian Johnson, and the way it portrays a heightened and dramatic story about high school kids and how that perfectly captures how I felt in high school. At that point in my life, I had never experienced anything more adult or stressful than that. So everything at that point felt really severe and scary and dramatic. The story in Brick is totally unlike anything that would ever actually happen in a high school, but I'm able to project myself in to it because I remember what it felt like to take things that seriously.
I think it just comes down to how you relate to the media you consume. I bet a lot of people do it without even thinking about it.