It was a 2D platformer, made sometime in the 2010s.
The main character was a penguin, who used an orange sword and a blue cybernetic boomerang as weapons.
In the description, the game was called "retro" or "retro-styled", and it looked like a cross between Mega Man X and Final Ninja Zero by Nitrome.
The game I'm looking for was probably not made by Nitrome.
The first level/tutorial was called "Like You Just Woke Up".
I will have to check this out on my computer; there were some legitimately good Flash games over the years. Hello Worlds! is one and I see it listed on this website.
I'm glad I was able to help you rediscover a flash game🤗