A classic application of the Mellin transform is in radar engineering. Recall that the resolution in time of a signal is the reciprocal of the spread of its Fourier transform. In radar (or sonar) a signal is reflected from a moving target and you would like to accurately determine its velocity. The velocity resolution of the signal is the reciprocal of the spread of its Mellin transform. This application is worked out in:

<A HREF="http://www.norbertwiener.umd.edu/crowds/documents/Ovarlez93.pdf">Cramer Rao bound computation for velocity estimation in the broad-band case using the Mellin transform</A>

To develop the physical intuition you are after, this article might be informative:

<A HREF="http://scitation.aip.org/content/aip/journal/jmp/15/6/10.1063/1.1666723">The power spectrum of the Mellin transformation with applications to scaling of physical quantities</A>

> The Mellin transform is used to diagonalize the dilation operator in a
> manner analogous to the use of the Fourier transform to diagonalize
> the translation operator. A power spectrum is also introduced for the
> Mellin transform which is analogous to that used for the Fourier
> transform. Unlike the case for the power spectrum of the Fourier
> transform where sharp peaks correspond to periodicities in
> translation, the peaks in the power spectrum of the Mellin transform
> correspond to periodicities in magnification.