In some cases, a CPU, once removed, can reveal what type of system they were used in, because of certain business practices (effectively bricking CPUs so they can't be used in the competitor's server).
I know of a single real-life example currently in the market; the AMD PSB vendor-lock-in function on its Epyc (Rome and later) and Threadripper Pro (gen 3 and up) CPUs.
These are fuses that are burnt, so the change in state is one-way only.
Now suppose that the computer vendor that sold that chip is, backed by its state, a security adversary. The computer vendor could use a special pattern, unique to the motherboard, to identify it exactly.
Or, more nefariously, once the target's machines are compromised, a hacker could use this mechanism on each one of them to assign the computer's CPU fuse code to one matching an unknown private key on boot, effectively permanently turning it into a brick (as it will refuse to decrypt the firmware). That could cause significant damage and downtime, greater than a security failure where the machines would simply be flashed back to a known good state.