Almost all registers are implement as so called D-Flip-Flop.
The ones in the System Control Block as well as the ones that make up the peripherals, timers, infrastructure etc.
Some "register blocks" are implemented as synchronous memory. This is the case if you need the result of one or maximum two at a time.
If you need to read only one of many registers they use single ported synchronous memories.
If you need to read maximum two of many registers they use dual ported synchronous memories.
configuration / communication memory that can be addressed by bus-systems like i2c or modbus is referred to as registers, are these also actual registers?
To 'store' (or write) a few bits, a register is the most suited structure. Only if you have many, many bits one tends to use memories of the various kinds, SRAM, EEPROM, FLASH, DDR.
If we talk about 'status' registers you get into less defined territory. e.g. A 'data available bit' can come from comparing a FIFO level with zero. But an 'overflow' is likely a real register which gets set when data arrives with a full FIFO.
are actually implemented by some kind of bistable circuitry like e.g. general purpose registers are?
I assume you mean with a "bistable circuit" a set-reset flipflop. or some other configuration with cross-connected NAND or NOR gates.
The answer is NO.
Registers are implemented as a pair of front-to-back connected inverters: Transmission Gate based D Flip Flop. This link has great pictures of how they work. These are used for nearly all registers, also for what you call "general purpose" registers. In fact in a chip there are no "general purpose registers". All registers are equal.