Yes, there are a fixed number of registers.
When the OS switches from 1 thread to another, it copies the state of the finishing thread, including the current values of all the registers into its descriptor, then copies the values from the the descriptor of another thread into the registers.
These descriptors are stored in RAM, although this is in part what Hyper Threading set out to fix, allowing 2 threads to be stored in a single CPU core at a time, but that is an entirely different discussion.
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