I'm not sure on the timeline and models, but I think by 60s all P basses had .047 (.05) uF caps. Some older models I think had .1 uF, which would also work - it just goes deeper into the lows as you roll off the highs.
Note that it doesn't really matter all that much what value cap you have until the last ~25% of the tone pot turn. Most of the "tone" effect before that last bit seems to come from changing the resistance (i.e. the pot itself). So unless you often roll off your tone by 75% or more, I wouldn't be too concerned about it.
Different brands and materials don't affect the sound (IMO, IME, this is a controversial topic), except that with looser tolerances the actual capacitance may be +/-20% (or more) from the nominal value. I think if you get some poly/film or even ceramic caps with 5-10% tolerance, you should be just fine. They're cheap, so try out a bunch (with alligator clips you don't have to solder), see what you like.
YES IT DOES MATTER ! The change in value determines the "Hinge Point" where the roll-off starts! The larger the value, the lower the frequency where the roll- off starts. 1/2 value = 2 x frequency. For example a 0.1 uF cap might roll off frequencies starting at 1 kHz, where a 0.05 cap would not start working until 2 kHz. ( frequencies shown are only for demonstration). THAT is significant.
Also, by starting at a lower frequency the larger value gives a deeper cut at the high end , since the rate of cut is the same for all caps in this type of circuit.
However, he is right that the cap material is not significant AT ALL.
Sorry, this is just fun.