What happens if you have a “race condition” on the lock itself?

For example, two different threads, perhaps in the same application, but running on different processors, try to acquire a lock at the exact same time. What happens then? Is it impossible, or just plain unlikely?

Have a “race condition” on the lock itself is impossible. It can be implemented in different ways, e.g., via the Compare-and-swap where the hardware guarantees sequential execution. It can get a bit complicated in presence of multiple cores or even multiple sockets and needs a complicated protocol (MESI protocol) between the cores, but this is all taken care of in hardware.

Compare-and-swap (CAS) is an atomic instruction used in multithreading to achieve synchronization. It compares the contents of a memory location with a given value and, only if they are the same, modifies the contents of that memory location to a new given value. This is done as a single atomic operation. The atomicity guarantees that the new value is calculated based on up-to-date information; if the value had been updated by another thread in the meantime, the write would fail.