If this issue is specific to the analogue sticks and not any of the tilt/gyro sensors, i can think of several strategies to combat this.
I would be interested to know what transducers are being used to detect movement. Potentiometers, lasers, hall effects, ultrasonics are all potential sensing techniques that can be used to measure position. But my guess here would be that its most likely a pot, but could very well be a laser sensor of one kind or another.
If they have used potentiometers my guess is its from heating causing changes in component resistance creating zero point shift.
If you could design the mechanism well, a superior analogue input would be a high res counting wheel set up for each direction of travel. No drift, sub millimetre precision, probably too expensive and too big though.
There are numerous ways of re calibrating sensors too which could be used if a user identifies the problem and was willing to go through a couple of steps of wiggling to troubleshoot. This is probably the easiest strategy to implement but requires provision in the firmware of the controllers and the OS of the console.