My guess is that it is factory original. The logo looks right.
Here's my thinking:
In the early '60s Fender basses were available with neck width options. Though very rare, the most common option was for a P Bass with a narrow Jazz width neck. At the time they simply put a Jazz neck on a P and put a P Bass logo on it. No one was any the wiser as the necks were interchangeable.
The problem came when Fender started to put binding and then blocks and binding on their jazzes in '65/66. A narrow neck option on a P now meant getting these features. Fender eventually realised this anomaly and started to make bespoke narrow width P Bass necks, but this didn't happen till around '68.
I have seen a couple of early '66 P Basses with bound, narrow necks, almost certainly because of this use of a stock jazz neck. There is also a well known Charlcoal Frost Precision from '68 which is not only block and bound but has a maple cap neck! It's appeared in a few books (most recently Fender, the golden years) and almost certainly was the result of an order for a narrow (maple) neck, which resulted in the factory putting the order through as if it were for a Jazz and not a Precision. I wonder if the owner was shocked to see the blocks and binding!
Johnny