In my experience as someone who made the transition from engineer to PM and have seen many others do it, an "official" transition within a company is hard-to-impossible if your company doesn't explicitly have a track for that. (Megacorp tech companies often do, mid-sized and below often don't.) If you're a good engineer, they don't want to give up your proven skills for an unproven lateral move.

What I've seen happen successfully time and time again is an "unofficial" transition that then turns "official" when you change companies. The way it always works is to find a project that doesn't have a PM for whatever reason but could really use one, and slowly become the "effective" unofficial PM once you join the team by taking on PM tasks that are blocking the team. Your manager's probably not gonna say no to things that urgently need doing but there isn't approval for an official PM.

Then start interviewing at other companies for PM roles and explain "well I was never an official PM but I did all the PM work like x, y, z". They don't care what your title is, PM's are known for transitioning from every other job under the sun.

I mean, that's how a lot of promotions work anyways -- first you demonstrate you're already performing at the next level, then you get it officially. It's just that the eng-to-PM transition isn't recognized officially at a lot of places, so you need to do it cross-company.