I am a first time DIY on a shower handle. I removed our old shower handle it does not have a brand name on it. The way it operates is that you pull knob out to turn water on. I have tried a trim kit for a delta and a moen and neither seem to work with this type of valve.
That's definitely a Moen. It has a "pull out dial" handle or (more specifically) Moen 96797 Chateau One-Handle Tub and Shower Knob; and I think the cartridge is 1225B.
Here's a video about replacing the cartridge (it shows exactly the save valve) and in the beginning you can see the tear drop shaped handle.