The version I played was the CDX, which bizarrely you could put batteries in and carry around like a Discman: https://retroconsoles.fandom.com/wiki/Sega_CDX/Multi-Mega
IIRC it did not officially support the 32X, but it did work?
The Genesis started losing to the SNES. They thought they could improve/extend it with the Sega CD but that flopped.
Nintendo of America realized they needed something better. So they designed the 32X. They later planned to sell a Genesis with one built in, sort of like the CDX. Honestly 32X + Sega CD was pretty good.
But Sega is a Japanese company. And Sega of Japan didn’t care for what Sega of America was doing (and possibly wanted to teach them a lesson?). So they basically refused to support the 32X, leading to very few games.
To make things worse Sega of Japan had their own solution to the problem, the Saturn. It was further off (this why the 32X made it out), but it got announced. And it was incompatible. Why buy a 32X if it’s going to be obsolete in like a year anyway?
So SoJ basically undermined/killed the 32X, though it may not have been a great idea in the first place.
Sega found out about Sony and got scared. I believe that’s why the added the second CPU to the Saturn (making it harder to program/more expensive). But they decided to beat everyone to market to see things up with a first mover advantage. So they announced the console officially and said “It’s out today in the US”. That was a huge bombshell no one expected.
Not even KB Toys and Toys R Us. They didn’t know and were PISSED. They were responsible for the vast majority of game sales. KB pulled all Sega stuff and stopped selling it. Huge loss. TRU did something too (lack of advertising? Low stock?).
Between that, the insane price of $400, and the lack of promotion (it was a surprise launch so no pre-release coverage) the Saturn started in serious trouble and never recovered.
Sony, famously, went up to the mic at their press conference, said “$299”, and walked off stage. It was a huge hit.
Sony had money (Sega wasn’t doing hot), partners (Sega pissed ‘em all off), a lower price, and Nintendo didn’t come out until later. Nintendo was also expensive to develop (carts) and didn’t have much space (carts).
So Sony cleaned everyone’s clocks.
(I love video game history. I got to live through such an amazing time)
But it was sort of odd. The 32X didn’t seem too stable since it stuck up so far. I think they later had a piece of plastic they offered (in box or via mail?) to fix that.