The clarinet and the soprano sax are similar in size, but the clarinet is able to play much lower notes than the soprano sax. Why is this the case? I understand that when the clarinet is overblown the pitch goes up by a 12th, not an octave, which extends the range, but I don't understand the physics of how it can play so much lower than the soprano sax. It seems like it could be partly to do with the size of the mouthpiece (soprano sax has a much smaller mouthpiece) or possibly to do with the bore shape, but I'm not really sure whether any of these reasons are right.
Both instruments (as well as all single- and double-reeded instruments and all brass instruments) behave as tubes closed at one end. The difference is that saxophone (and all other reeds) are conical, whereas the clarinet is cylindrical.
For a conical tube closed at one end, the fundamental wavelength is twice the length of the tube (Actually, it's a bit more complicated than that, but that will work for a basic understanding).
For a cylindrical tube closed at one end, the fundamental wavelength is four times the length of the tube (and the instrument only resonates odd harmonics). So for a basic approximation, a clarinet is going to be about an octave lower than a similarly sized conical instrument.
You pointed out the reason. The clarinet acts as a stopped pipe. An open pipe has a node (little air movement) at each end and an anti-node (lots of air movement) at the middle (or maybe I've got this backwards; check the physics of wind instruments pages.) Thus the clarinet's fundamental wavelength is twice the pipe length and the saxophone's is equal to its pipe length. Thus for the same length of pipe (the effective length of the instrument), the clarinet would sound an octave lower than the sax.
Next, (as mentioned in the question) the clarinet's natural overtones are the fundamental, third overtone, fifth, etc., while the sax's are the fundamental, second,third, etc. The clarinet starts and octave lower and its natural vibrations are the odd harmonics. A sax's natural vibrations are all the harmonics.