Why not the piano as a reference for bass

I think that the "attack" has a lot to do with it.  Whereas a bass is "plucked" by hand or played with a pick, a piano is struck with a felt-covered hammer and this presents a completely different "leading edge."

Actually, no..., Ray played a Fender Rhodes Piano Bass with his left hand.  He never played pedals.

In several of their tracks, he is overdubbed by a bass guitar.

https://youtu.be/Fiq9VDkGVnI

Played clarinet from 5th through 12th and keyboards in a garage band in the late 60s early 70s.

Had equipment to die for back then (Hammond, Hohner, Vox, Farfisa, Moog and EML synths), but sold it off to pay for college. 

Another thought on the topic.  You hear basses go down to their low E-string literally all the time, so people have a feel for it (did someone say reference?).

But how often you you hear a piano playing low A?

As to the Doors, I would hope that one's system allows them to differentiate the sound of a bass guitar...,

https://youtu.be/ptA39Awo0FE

and a Rhodes Piano bass...,

https://youtu.be/nOJSmXSFCWk

It should also be obvious as to whether Ray is playing a Hammond, a Vox, or a Gibson organ in each cut.

@richardbrand 

https://youtu.be/XBXUP5GqYJs

Synthesizer beats an organ for bass.

All organ bass has a slow attack compared to what can be done with a synthesizer.  And the right synthesizers can go infrasonic, and not just to modulate the audio oscillators, filters, and voltage controlled amplifiers.

The attack on a pipe organ on a really deep note takes time for the pipe/chamber to fill so the note comes on more gradually.

A synthesizer can play basically any waveform all the way down to subharmonic.  A subharmonic square wave with an instantaneous rise time is harder on the speakers that have to make the transition more quickly. 

It's not limited to square waves.  A sawtooth wave also has an instantaneous rise or fall time, as the case may be.

In the real world of music, a square wave would most sound like a clarinet, a sawtooth wave like a string, with bow spikes, a sine wave like a flute, and a triangular wave "flute-like" though none of these instruments are "pure of wave".

So, what's Joni playing (make and model)?

https://youtu.be/_kyQEa-QUd8