What is "shake-induced voltage"? Is that anything to do with the parts inside the tube literally moving and inducing current?
What makes a 12AX7 different than a 12AU7? Why does one amplify more than the other?
Let's ignore the different current and power ratings for the tubes. The 12AU7 has a mu of about 17, the 12AX7 about 100. Mu, or amplification factor is not constant, but let's pretend it is since it is more constant than other tube parameters.
The control grid is the element that determines how much plate current flows in a triode if plate to cathode voltage is held constant. The 12AX7 amplifies more than the 12AU7, so maybe there is something different about their grids. In fact, there is.
The control grid is a spiral of wire wound around 2 support rods, and physically between the cathode and plate. If you apply a voltage to the grid that is negative compared to the cathode, it tends to electrostatically repel the electron flow back towards the cathode. In other words, more negative g1 voltage lowers plate current. Indirectly, if the grid becomes more effective at doing this, meaning a bigger current change is caused with the same change of g1 voltage, then you can wind up with more amplification after inserting a plate load resistor.
How do you make the grid more effective at controlling plate current? How about winding that spiral tighter and tighter, so that the wires are closer together? Now the electrostatic force acts over a smaller physical space, because the wires are closer together, so a smaller voltage will achieve the same effective control of plate current, or it is more effective with the same g1 voltage applied.
This is great if those wires are perfectly immobile, but what happens if there is slop and the wires can move? Then an outside force shaking the tube can cause more and less amplification to occur. Viewed from outside the tube, you can't really see that the internal structure is moving, you just know that when you use physical force to shake the tube, or when sound waves hit it and shake it, there is a change in output of the tube that is directly related to the force applied. From your frame of reference outside the tube, this all looks like a change you cause by shaking, or a "shake induced voltage".
Tube geometry dictates many of the tube's parameters; distance from the grid to the cathode, distance between grid wires (winding pitch), distance from cathode to plate, distance from grid to plate, size and weight of the plate material, and many more. Those mica spacers at the top and bottom of the tube structure do a number of things, but the first and foremost is to attempt to hold the entire tube system rigidly in place and prevent the tube elements from shifting and changing the tube's performance.
But maybe the holes cut in the micas are just a tiny, tiny bit too big. Now the tube structure has some slop, and any force that might act to move the innards will cause a momentary change in characteristics.
But spacer holes could be too big, the spacer could be cut just a hair too small to hold against the inside of the tube envelope, the outer envelope could be just a bit too large, grid wires could be wound just a little too loose, support rods could be just a bit too small, plate metal or the cathode could be just a little too flexible and move in mid-span (changing the distance between elements), and on and on.
There might be no such thing as a microphonic-free tube; but there are tubes that have so many design and construction features/considerations that the level of microphonics is too low to affect the circuit they will be used in.
There is an audiophile tube vendor that has high prices, but may very well be on the up-and-up. He sells preamp tubes in at least 4 different grades, based on measured noise and microphonics. The poorest performers are graded for line-stage use, because while they do perform well, there is some noise or microphonic behavior, but the tube is intended to be used in a circuit where the signal level will be so much higher than the offending behavior as to mask it. Used in accordance with his grading system, there will be no perceived defect. His highest grade of tube is for use in extremely low-level phono stages, where even the slightest hint of noise or microphonic behavior will be very large in comparison with the intended signal. Tubes meeting this criteria are very hard to come by, and his prices reflect that.