The best action games like Golden Axe are Streets of Rage 4, Streets of Rage 2 and Dragon's Crown. Below are 6, each with the reason it is here — and, at the end, what filtering to action costs you.
Filtering to action means you want the fighting to carry the game rather than the story or the map. It is the right filter when what you actually miss is how something felt in your hands.
What you get: Combat you get better at. Action games put their design budget into the moment-to-moment, so the thing you are doing in minute two is the thing you are still doing in hour twenty, only better.
What made Golden Axe worth playing in the first place is covered on the full list. This page is the action part of that answer, and only the action part.
The best modern beat-em-up, and it honours the Sega originals directly.
The Genesis high point of the genre.
Fantasy brawling with RPG depth and gorgeous art.
Four-player fantasy brawling built for a sofa.
The arcade sequel, and far better than the console follow-ups.
Top-down arcade fantasy action, explicitly retro.
Pacing and world. Action games rarely have the quiet stretches or the writing that the adventure and RPG picks bring, and a few of them are shorter than they look.
Specifically, these are strong answers for Golden Axe that are not action games, so they are not on this page:
The full list is on games like Golden Axe — 9 picks, no genre filter.
Streets of Rage 4. The best modern beat-em-up, and it honours the Sega originals directly.
No. Golden Axe released on the Sega Genesis, but a recommendation is only useful if you can play it, so the picks above span platforms and eras. What they share is the genre and the reason.
A small set of verbs and a game that keeps finding new questions to ask of them. The weak ones add buttons instead; the strong ones make you better at the ones you already have.
No, and difficulty is a poor proxy for depth. Some of the hardest action games have very few decisions in them, and some of the deepest have a difficulty setting that lets you meet them halfway.
By hand. Each pick was written for the full Golden Axe list first, on what the game actually does, and this page is the part of that list that is action — never a genre tag lookup, which is how "best racing games like Super Mario 64" used to be a page on this site.
Or read the action guide for where to start in the genre generally.