I will use the TM4C123 Arm Microcontroller/Board as an example.
Most of it's I/O registers are memory mapped so you can get/set their values using regular memory load/store instructions.
My questions is, is there some type of register outside of cpu somewhere on the bus which is mapped to memory and you read/write to it using the memory region essentially having duplicate values one on the register and on memory, or the memory IS the register itself?
There are many buses even in an MCU. Bus after bus after bus splitting off like branches in a tree. (sometimes even merging unlike a tree).
It may predate the intel/motorola battle but certainly in that time frame you had segmented vs flat addressing and you had I/O mapped I/O vs memory mapped I/O, since motorola (and others) did not have a separate I/O bus (well one extra...address...signal).
Look at the arm architecture documents and the chip documentation (arm makes IP not chips). You have load and store instructions that operate on addresses. The documentation for the chip (and to some extent ARM provides rules for the cortex-m address space) provides a long list of addresses for things. As a programmer you simply line up the address you do loads and stores with and the right instructions.
Someones marketing may still carry about terms like memory mapped I/O, because intel x86 still exists (how????), some folks will continue to carry those terms. As a programmer, they are number one just bits that go into registers, and for single instructions here and there those bits are addresses. If you want to add adjectives to that, go for it.
If the address you are using based on the chip and core documentation is pointing at an sram, then that is a read or write of memory. If it is a flash address, then that is the flash. The uart, the uart. timer 5, then timer 5 control and status registers. Etc...
There are addresses in these mcus that point at two or three things, but not at the same time. (address 0x00000000 and some number of kbytes after that). But, again, not at the same time. And this overlap at least in many of these cortex-m mcus, these special address spaces are not going to overlap "memory" and peripherals (I/O). But instead places where you can boot the chip and run some code. With these cortex-ms I do not think you can even use the sort of mmu to mix these spaces. Yes definitely in full sized arms and other architectures you can use a fully blow mcu to mix up the address spaces and have a virtual address space that lands on a physical address space of a different type.