What's the meaning of a Bode Plot when talking about nonperiodic inputs?
Bode Diagram characterizes a system, regardless of the excitation signal.
The transfer function of a system, is valid for any signal. Bode Diagram is another way to visualize the transfer function of a system.

An aperiodic signal, also has representation in the frequency domain. An aperiodic signal, can also be considered as the sum of sinusoidal components, although its spectrum is not "constant" as in the case of a periodic signal.

We can consider that an aperiodic signal has a spectrum that varies every moment in its composition. At one time, this spectrum has a specific distribution of components; at that moment, for the aperiodic signal applied to the system in question, the spectrum will be modified according to the frequency response of the system.

Accordingly, the aperiodic signal will be affected by the frequency response of the system to a greater or lesser extent, according to their instantaneous spectral composition.

For a basic relationship between the frequency response and an aperiodic signal, look at the step response and its relationship to the frequency response.

Example

Consider the following aperiodic signal



wich has this spectrum



Of course, this spectrum is continuous, as for an aperiodic signal. This signal is the input signal for a system with this transfer function

\$
H(z) = \dfrac{0.2\,z + 1}{0.5\,z^2+1.5\,z}
\$

and this Bode Plot



As you can see, it is a high-pass filter.
The output signal looks like



and has this spectrum



Note that high frequencies have greater amplitude than the same frequencies in the input signal, as befits a high-pass filter.