What's the maximum memory address space that the processor can access directly if it connected to a 16-bit memory?
I think there is a confusion here between the number of memory addresses and the total size of memory that can be addressed.

The microprocessor can access $2^{16} = 65536$ memory addresses, regardless of the size of data held at each memory address.

If each memory address holds one byte ($8$ bits) of data then the total memory size that can be addressed directly is $2^{16} \text{ bytes}$, which is $2^6 \text{ KiB} = 64 \text{ KiB}$.

On the other hand, if each memory address holds two bytes ($16$ bits) then the total memory size that can be addressed directly is $2 \times 2^6 \text{ KiB} = 128 \text{ KiB}$. 

It also appears your text is using KB as an abbreviation for KiB, which is a common shorthand but not strictly correct.