The answer to your question is entirely system specific. However, most processors know nothing about threads. They only support processes. Threads are generally implemented by created separate processes that share the same address space.
When you do a system service call to get a thread ID it is going to be implemented in the same general fashion as system service to get the process id. Imagine how a get process ID function could work in a system that does not support threads. And to keep it simple, let's assume a single processor.
You are going to have some kind of data structure to represent the current process and the kernel is going to have some means of identifying the current process (e.g. a pointer in the kernel address space to that process). On some processors there is a current task register that points to a structure defined by the processor specification. An operating system can usually add its own data to the end of this structure.
So now I want to upgrade this operating system to support threads. To that I must have a data structure that describes the thread. In that structures I have a pointer to a structure that defines the process.
Then get thread ID works the same way get process ID worked before. But now Get Process ID has an additional step that I have to translate the thread to the process to get its id (which may even be included in the thread block).