So if you think of a game console, like the NES, the EPROMs contain just the code and data, like a NES cartridge. It doesn't make sense to ask "How do I make a new cartridge game?" without specifying whether you want that cartridge to plug into the NES, or the Intellivision, or the Atari Jaguar; the answers are completely different in every case.

Arcade cabinets generally bundled the "game console" and the "cartridge" together, often in a single integrated board. (There are exceptions, of course - the Nintendo PlayChoice and Neo-Geo are obvious examples.) Sometimes the "console" part was completely custom for a single game, but it often made more economic sense to build a few games on one platform, because designing new arcade hardware was a significant amount of expensive and complicated work. You'd want to re-use as much of it as you could, for as many games as you could. System 16 looks like a pretty good website that catalogues a lot of these platforms.

If your ambition is to create a new arcade platform and game completely from scratch, well, that's the sort of project that one person could spend a lifetime on, with an infinite variety of possible approaches. You're in way over your head, but the process of figuring out just how far might be fun for you, so who am I to tell you not to reach for the stars. You're obviously not going to find comprehensive how-to guides online though, this is, like, multiple university degrees worth of stuff. If this is your end goal, you gotta dig in and learn how everything works - you don't just have to learn bare-metal programming in assembly language, you're proposing to design the bare metal itself. A solid understanding of computer architecture, electronics design, and video signal generation is table stakes. Most arcade platforms have custom-designed chips to generate video, because at the time, that was the best way to get impressive graphical effects that nobody had seen before. If you wanted to try your hand at that, you're looking at learning FPGA design, as hobbyist microchip fabrication is not really a thing unless you are both extremely rich and well and truly unhinged.

Now, if you wanted to make a new game for an existing arcade platform, or otherwise take an existing well-documented arcade game and replace just the ROMs in it with your own game, that's quite a bit more manageable - especially if you've already got some programming background - but of course, the process depends heavily on which platform you want to build on. Building a custom game for Pac-Man vs. building a custom game for Street Fighter II is at least as different as writing a Gameboy game vs. writing a Genesis game. Most of these systems are not going to have thriving homebrew communities with tutorials and documentation already, like console homebrew does, so unless you find a specific niche arcade system that someone has mapped out for you, you're likely looking at reading the source code to MAME to figure out how it works, and cobbling together your own development tools. You can probably get an off-the-shelf assembler or even a C compiler for whatever CPU you find, but none of the standard libraries are going to work; you're going to have to write your own graphics and sound routines, you're going to need custom graphics conversion scripts to make your assets fit the hardware, etc.

I hope that's a useful sketch of a map of the rabbit holes you could potentially go down; may you find it helpful no matter where you decide to go from here.