Games weren't getting individual custom CPUs or anything that exotic, but from roughly 1980-2000 the industry was still riding the peak of Moore's Law, where chips were getting faster and cheaper seemingly every day, and arcade developers were constantly trying to leapfrog each other. Each game system really only needed to run a single game, so there was no real barrier to having unique boards per game -- if a particular game needed a few extra RAM chips, or the CPU maybe needed a higher clock, there wasn't much barrier to that.
NOTE: System16 is a treasure trove of information in general, but it looks like their main landing page at https://www.system16.com has some kind of hijack ad situation going on? The links below seem unaffected.
Konami was kind of the poster child for bespoke hardware; they're famous/infamous for seemingly every single game having some kind of weirdo bespoke hardware iteration: https://www.system16.com/museum.php?id=5
Sega had a more typical approach. They would typically have a minimum of several games per arcade system, and sometimes dozens: https://www.system16.com/museum.php?id=1
But, even then, there are lots of variations to the rule. Taito operated more or less like Sega, but (to choose one at random) you might see things like this where one game on this particular board (Cadash) was equipped with a faster CPU than the others: https://www.system16.com/hardware.php?id=652
There are some fun exceptions to what I've written though: truly one-off hardware.
it runs through a variety of Sega Model 2 arcade system games. theoretically there were about a dozen "Model 2" games.
buuuuut, as the video details, the "Model 2" moniker refers to an entire generation of Sega arcade boards and there were significant hardware upgrades and new chips added throughout the lifetime of Model 2. with the result actually being something close to bespoke per-game hardware.
also there's this hilarious (to me at least, from an engineering perspective) look at the extremely unique Popeye arcade hardware, with three different graphics chips each drawing different layers on the screen. that was some bespoke-ass hardware
NOTE: System16 is a treasure trove of information in general, but it looks like their main landing page at https://www.system16.com has some kind of hijack ad situation going on? The links below seem unaffected.
Konami was kind of the poster child for bespoke hardware; they're famous/infamous for seemingly every single game having some kind of weirdo bespoke hardware iteration: https://www.system16.com/museum.php?id=5
Sega had a more typical approach. They would typically have a minimum of several games per arcade system, and sometimes dozens: https://www.system16.com/museum.php?id=1
But, even then, there are lots of variations to the rule. Taito operated more or less like Sega, but (to choose one at random) you might see things like this where one game on this particular board (Cadash) was equipped with a faster CPU than the others: https://www.system16.com/hardware.php?id=652
There are some fun exceptions to what I've written though: truly one-off hardware.
Namco's Thunder Ceptor, with its mode7-ish backgrounds... in 1986! https://www.system16.com/hardware.php?id=523
Gaplus/Galaga3 had its own board for some reason: https://www.system16.com/hardware.php?id=518
Pole Position I and II (essentially the same game) had its own hardware: https://www.system16.com/hardware.php?id=515
I think a lot of Midway's late-80s/early-90s boards (let's say, NARC through MK) tended to evolve from game to game.