I think what everyone has said above is very valid. About playability, simplistic but versatile design, tonal strength etc., but there is one point I've been thinking about for some time now that hasn't been mentioned.
Let's first start with another question: Why does the Philips head screw driver work so well, in so many situations? The answer is two fold, firstly it is a great design which is sturdy and useful and fulfils all the requirements of a screw driver.
The second is that since the Phillip head screwdriver was decided upon as the standard, the entire screw driver world has accommodated around the screw driver itself.
In the same way, Leo Fender did a great job designing a bass. While it's clearly not subjectively the best bass in the world for everyone, many people liked the tone and started using them in bands and studios. Here's where the game changes:
Since everyone is using the Fender Jazz Bass, bands and musicians play in a way which complements that particular bass tone. The type of drum kits, and how the drums (particularly the kick) are tuned and played in relation to the jazz bass tone. An engineer at every pro recording studio around the world needs to learn how to mix a band with a fender bass in it, taking into account the strong mid frequencies that it presents, in comparison to the P bass or the double bass used previously.
So what we end up with (sonically and aesthetically) in many many bands / genres and recordings around the world is a fender jazz bass sized whole to fill.
Thoughts?