I've been thinking a lot about this lately.
If the familiar drop to zero hit points, it is sent back to the pocket dimension, and the spell must be recast. When recast you get the same spirit back again.
That isn't the only death a familiar can face.
Power Word Kill for instance says:
If the creature you chose has 100 hit points or fewer, it dies.
So, if a character spent a 9th level spell slot, they could kill the familiar dead. Certain Celestial and Fiends when killed reform on the celestial plane, abyss, or the 9 Hells (like Devils, Demons, etc.). So, death might not be the end of them. The Fey, however, is likely toast.
Recasting the spell after that death will still work, but you're likely just going to get a new spirit.
And Disintegrate says:
On a failed save, the target takes 10d6 + 40 force damage. If this damage reduces the target to 0 hit points, it is disintegrated.
It is unclear which is more specific, and which happens: the Familiar rule or the Disintegrate rule.