When a device is assigned an IRQ, does that mean that it has sole access to certain registers associated with that IRQ?
IRQs and Registers are totally different things.
An IRQ is an message mechanism and a register a storage.
The internal registers of the CPU are not accessible from outside the CPU.
The internal registers of a hardware device may be accessible by the CPU - then these are known as "ports".
However, nowadays the 'ports' also often are mapped to an internal controller that considers it as input data that may be processed in different ways (e.g. storing in an internal register, processing it to create output data, storing it in memory or handle it as "event" for a state machine).