Why does thrust-to-weight ratio depend from max Mach number? Why, for jet airplane design, does maximum Mach (or maximum speed) affect thrust-to-weight ratio so strictly? (as you can see for example in Daniel P. Raymer "Aircraft Design: A conceptual Approach" Table 5.3 page 80, where you can find $\frac{T}{W_o}=a\cdot M_{max}^C$, where “$a$” and “$C$” are empirical factors, for jet transport $a=0.267$ and $C=0.363$)

Just now I did notice to detail that Raymer say : "The equations were developed by the author using data from Ref 1" Ref1: i.e. the famous annual book : "Jane's All the world Aircrafts" . But anyway "maximum speed" (or maximum Mach number) in this case the maximum speed is ...the speed that the aircraft is not able to exceed  or.... the speed that is forbidden for the pilot to try to exceed because it can be dangerous? Because, if it's the first case it's more understandable; in fact, if the other parameters are fixed, with the same fixed distance, if T/W is low the acceleration is lower and so before the end of the distance, you can arrive to low speed. At the opposite if T/W is high the acceleration is higher and so before the end of the distance, you can arrive to high speed. What do you think? So, in Janes book, what is the definition of Maximum Mach number?
Disclaimer: I have no copy of Raymer's book, so I have to guess what this table looks like.

There should be no direct dependency, except that all aircraft need to operate in the real world. If you look at the wing loading of regular jet transports, you will notice that they bunch together around 500 - 600 kg/m². This is the maximum wing loading which still allows them to operate from normal airports - go to higher values an you need a dry lakebed for take-offs and especially landings.

To push the resulting wing to higher flight speeds needs more thrust - hence the dependency. If you start with a higher wing loading (say, 800 kg/m²), the thrust/weight ratio for reaching the same Mach number is lower. The approach and landing speeds grow with wing loading as well, which in turn affects the resulting landing distance. I guess there is a maximum landing distance requirement for all aircraft in the table which is not explicitly mentioned.