Control theory: is there any actual application for D in ABCD matrix?

A lag-lead compensator in state-space form will have non-zero \$D\$.

In \$H_2\$ or \$H_{\infty}\$ control design if the performance variables include the input (e.g. keep it 'small') then again \$D\$ will be non-zero.

They can also appear when a continuous-time system is approximated as a discrete-time system (e.g. convert \$\frac{1}{s+p}\$ using Tustin's transform).

That is a feedforward term. As long as the plant model is understood & accurately captured within a controller, a feedforward term can provide a performance boost that can mitigate aspects of a PID controller.

Take for instance a controller to regulate the speed of a car. If correctly modeled, a given amount of accelerator-pedal would be needed for a given speed to overcome drag. Rather than waiting on a closed-loop PID to increase the accelerator force to overcome the present drag + acceleration, the present drag component can be used as a feedforward term into the overall control topology.