Synthetik isn’t actually a Shmup, if you’re looking for games like it you want to look into twinstick shooters. A Shmup is an automatically scrolling shooter with where you are traveling in a single primary direction. A true Shmup gives you freedom to move all around the screen but the screen scrolls through the level at a fixed rate. Famous shmups include Gradius, R-Type, Raiden. There are a few different subgenres but the main offshoot you’ll see is Bullet Hell.
A great place to start is the Touhou games, and fan games (My favourite is Blue Devil in the Belveder). My all time favourite Shmup is Teslapunk, but it’s pretty obscure and not exactly popular.
If you want to learn more look up Shmup Junkie on YouTube and join his discord. Lots of great guys and a curated list of beginner games/goals for someone who’s interested in the genre. Trust me there is a Shmup for every taste out there. Try lots to see what clicks for you!