Using faster than light signals to synchronize clocks 
What would happen if one could communicate by non-luminous signals whose velocity of propagation differed from that of light? If, after having adjusted the watches by the optical procedure, we wished to verify the adjustment by the aid of these new signals, we should observe discrepancies which would render evident the common translation of the two stations. And are such signals inconceivable, if we admit with Laplace that universal gravitation is transmitted a million times more rapidly than light? [Poincare, 1904 address]

So to eliminate possibility of measurements by faster than light signals, Poincare assumed that the gravity should not propagate at faster than light speed. He limited it to the speed of light.
I think that Poincare is saying that if the clocks have been synchronized by light signals then using any faster-than-light signal would make it obvious that an inertial frame is moving. Assuming my interpretation is correct, why would it make it clear that inertial frame of reference is moving or why one "should observe discrepancies" if faster-than-light signals are used to synchronize the clocks? 
 A: 
Assuming my interpretation is correct, why would it make it clear that inertial frame of reference is moving or why one "should observe discrepancies" if faster-than-light signals are used to synchronize the clocks?

Your interpretation is correct, but in this case Poincare is wrong. Remember, if the date is correct then this address was given prior to Einstein’s paper On The Electrodynamics Of Moving Bodies. He would have been discussing Lorentz Aether Theory. 
If a signal propagated at v>c with respect to the Lotentz aether then it would be possible to detect the velocity of the aether using that signal. To make visualization easier let’s assume that the signal travels instantaneously with respect to the aether (similar conclusions follow for other propagation velocities). 
In that case, if you are at rest with respect to the aether and you synchronize your clocks optically and then emit this superluminal signal from midway between the synchronized clocks the signal will be received at the same time according to the optically synchronized clocks. On the other hand, if your clocks are optically synchronized while moving with respect to the aether then the clocks will disagree about the time that the superluminal signal is received due to the relativity of simultaneity for local time in Lorentz’s theory. 
Thus far Poincare is correct. A superluminal signal which traveled at a defined velocity with respect to the aether would violate the principle of relativity and allow the measurement of “absolute” velocity. 
However, where Poincare is incorrect is in thinking that a superluminal signal would necessarily travel at a fixed velocity with respect to the aether. Suppose, instead, that the superluminal signal travels at a fixed velocity with respect to the emitter. In that case, then the signals received at the optically synchronized clocks would be received simultaneously regardless of the velocity of the pair of clocks. 
Note, in this case, it is possible to show that a signal which travels at a fixed v>c with respect to the emitter can be used to violate causality (see https://pages.uoregon.edu/imamura/FPS/images/p263_1 ) whereas a signal which travels at a fixed v>c with respect to the aether cannot. Thus out of relativity, causality, and superluminal signals it is only possible to have two (see http://www.physicsmatt.com/blog/2016/8/25/why-ftl-implies-time-travel ). Most scientists believe that relativity and causality are correct so superluminal signals are impossible.
A: Note that although a super-luminal method would establish an objectively preferred frame for light - and light would become subject to the same interpretation as sound (which exhibits all the same properties as relativity, if the clocks themselves are all implemented through the exchange of sound signals) - the question could still remain as to whether this super-luminal method itself had any preferred frame, and therefore it would not intrinsically defeat those who argue that no such frame exists at the final frontier (although it would deal them a blow to credibility, and people would ask why an exceptionalist interpretation is always being applied to the fastest known method at any time).
If there is to be any experimental resolution of the question, then anyone postulating the existence of a preferred frame, must propose some technique by which it can be detected from within, rather than detected from outside by a hypothetical faster or instantaneous means of transmission.
