Plenty of people get into UX/UI with an ID background. Some designers like to focuses on the first half of the process other like the backend. I know quite a few designers that I graduated with that now focus solely in that area of design (front end). As for your “differentness” I would say 40% of my graduating class fell into the same boat. Most now are Design Researcher. Design Strategist or UX/UI hires. All with “traditonal” ID backgrounds.
But from what I remember a lot of them had similar issues. The solution for most was. Get experience anywhere. Sure they might have not landed superstar companies right out of school. But they got experience, learned the process and most are now making jumps to those larger/ more prestigious companies now. I cant help you with super specifics on transitioning into that role, but its not unheard of.