The professor is partially correct.
The data that is being saved when switching from user mode to kernel mode is the state of the CPU, not the process. User mode and kernel mode are CPU-related, not process-related. When the kernel executes some function on behalf of the user, the state of the CPU is saved beforehand. That is, the state of the CPU when it was executing the user mode code. Then, the kernel performs its action based on the user mode code's request. Finally, the kernel restores the state of the CPU back to the previously-saved state. The kernel may return the CPU back to user mode, depending on the kernel in question.
The kernel process has nothing to save, because it was not running beforehand. It (the kernel process) does a job on behalf of the user mode process, and then it is finished. The CPU can return to user mode, as far as the kernel is concerned.
Operating system components typically run in kernel mode, also called Ring 0 on x86 processors. User mode contains applications and is sometimes referred to as Ring 3.