The "return to center" behaviour of the N64 Control Stick is provided by a spring that pushes up against a plastic bowl which is underneath the stick. The stick itself slots through two curved plastic pieces for each axis, and the upward force basically forces the stick to 'center' at the lowest point of the bowl. The issue is as others have described. There actually never was any lubricant, so over time the bottom of the stick and the bowl just erode away. Eventually it reaches a point where the bowl is eroded away enough that the spring is no longer able to force the stick back to the center, and it just sort of flops in the dead space eroded in the bowl.
Nowadays, one can get replacement parts, and fairly easily restore/fix the problem. Add in a bit of PTFE lube or lithium grease and it greatly extends the lifespan of the replacement as well. Used to be able to get rather fancy steel bowls and analog sticks which significantly reduced the wear even more.
The more 'standard' thumbstick design has a similar self-centering mechanism. They are actually susceptible to the same problem. However, two things contribute to it being witnessed less frequently, I suspect. The first, is that the "bowl" in those sticks actually has lubricant which greatly reduces the wear, and the second is that the mechanical parts tend to outlast the potentiometers; that is, the sticks start to drift before you get the "floppy stick" problem to begin with and then you stop using it or replace the stick, and therefore it doesn't actually see enough wear to cause the floppy stick problem.