Does the impulse response of a system have any physical meaning?
The impulse response of an LTI (linear time invariant) system has the same physical meaning of the differential equation of a general dynamical system has.

So every output can be computed from the input and the impulse response convolution, without having to solve the differential equation. The frequency charactheristics can also be deduced from the Fourier transform of the impulse response.

Several properties of the (LTI) systems can be seen in the impulse reponse (and Fourier transform of it) expressions, such as slow system, fast system, wide band narrow band systems, overshoot, oscillations etc...