JR was one of the earliest manufacturers (if not first) to introduce the concept of modules on their radios. In the old days of AM/FM radios, the transmitter is built to transmit on a particular frequency band (e.g. 2.4GHz, or more likely 36MHz in those days). Then you could add the capability to work with a different frequency band by clipping a module into the back of your transmitter. As long as your transmitter (or module) and receiver were both AM or FM they'd talk quite happily.
When 2.4GHz came along the market became more fragmented, and every manufacturer came out with their own protocols, so you not only had to transmit 2.4GHz, but also speak the right protocol. This is why different brands of transmitter and receiver (e.g. Frsky/Taranis and Spektrum) can't be mixed interchangeably. Modules can solve that problem by giving your transmitter the capability to talk different protocols even though they might all be 2.4GHz (e.g. the Jumper Multiprotocol module).
These days the two most common use for modules is either support for new protocols (e.g. the multi protocol module on the Jumper radios, or a DSM module on Frsky). Or different bands, e.g. CrossFire which operates it's own protocol on the 900MHz band (for longer range).
The Frsky Taranis is probably the most popular radio today amongst quad pilots. When Frsky first made their radio they either bought the molds or licensed the design (can't remember which, but it was a legit "money changed hands" transaction, not a rip off clone job) for their case from JR. Once you know this, it's not at all surprising that a Frsky Taranis and a specific model of JR (I'm not sure which) are virtually indistinguishable when you put the side by side.
Since Frsky radios were made in the same moulds, it would be extremely unlikely for the Frsky module bay and JR bays not to be exactly the same size. You'll sometimes also hear them called "JR modules", or "JR module bay" for this reason. Whether the Frsky and JR modules are electrically compatible I don't know. i.e. a genuine JR module would fit in a Frsky bay, but I don't know if they'd work and not be damaged. I assume they would be electrically compatible, but I don't know of anyone who's tried it.
Other modern radios (e.g. the Jumper T16) have copied the design of the module bay so that popular modules (such as TBS CrossFire) also work on those radios. TBS don't sell different Crossfire modules for FrSky and Jumper radios, so these must be compatible with each other.
I'm not aware of any modules that aren't transceivers (either different bands, or different protocols). Hosting additional transceivers was the intended use of the module bay, I can't think of anything else I might want to install in the back of my transmitter that isn't a transceiver. Depending on what the pins do, and the voltage and current a module can draw from the radio, in theory there's no reason another device could draw power from the radio via the module pay.