The term "register" is vague, since there are two kinds of registers.
- Those who reside inside the CPU core, also referred to as CPU registers, data registers, accumulators and similar. These do not have an address.
- Memory-mapped hardware peripheral registers, used to control the various hardware built-in with the part. These have an address and are essentially a form of glorified RAM memory.
There aren't really any "microprocessors" as such nowadays, since pretty much every single part made is guaranteed to have lots of stuff on-chip. Interrupt/exception handling, MMU, oscillators, PLLs, cache memory, watch dogs, brown-out detection and so on.
The whole "microprocessor vs microcontroller" thing originates from the 1980s somewhere, when Motorola 68k was the hottest part out there. It isn't something you need to ponder nowadays. It rather seems like you found some outdated old book/teacher that stopped being accurate many decades ago.