You ask, for water containers across a variety of volumes,
"...the range of notes appears to be identical regardless of vessel shape. Why? And can you describe exactly what is making the sound?"
The short answer is they are definitely not identical. However, the container-dependent effects on the sound are small in comparison to the source-dependent effects.
It turns out that air bubbles are the primary source of sound in pouring water. In the freely available paper by Zheng and James 2004 entitled Harmonic Fluids, the authors state:
Ironically, the complex visible
motion of the air-fluid interface causes relatively little sound, in
part because visible surface motions are inefficient radiators of
sound waves at audible frequencies [Bragg 1920]. Instead, the
fluid shape vibrates harmonically at audio frequencies due to the
microscopic oscillations induced by internal air bubbles, and acts
like a shape-changing 3D loudspeaker.
The model, itself (which you can actually listen to examples of here) relies on a dual-domain Helmholtz Green's function, such that it includes two boundaries:
- the boundary of the liquid and it's container, $\Gamma_s$
- the boundary of the liquid and the air (where our ears are), $\Gamma_a$.
Both obviously depend on the form of the container! However, as I understand it, the size and number of bubbles has much more influence over the source strength and frequency of the pouring event.
One experiment you can do to see if this makes sense in practice is to change the way the stream enters the container (angle, height above fluid surface, diameter of the stream, etc.) Try to change the number or size of the bubbles. See how that changes the sound. Alternatively controlling for those factors with different containers may isolate the effect of the container's form.