Why are we using FFT for finding the periodicity of laser induced periodic surface structures?
Because the Fourier transforms for periodic signals are based on a basis of sinusoids. What better way to analyse a periodic signal than to represent it by harmonic waves. A bonus is that the DFT has a rather fast implementation in the various FFT techniques.
Another way to identify a period would be to form scalar products of the signal with shifted versions of itself. If you do it for many shifts using an equally spaced sequence of shifts, this amounts to convolution. Here again FT helps out, since after transformation, convolution turns into pointwise multiplication. The best match in the shifted scalar products corresponds to the highest absolute value in the magnitudes of the Fourier transform.
Be sure to apply sensible fade-in and out of the data segments to be analyzed. This is called windowing and prevents artefacts from not exactly capturing the period in selecting the data segment.
As Lutz said, the technique is used very widely in signal processing, which itself is needed in extremely diverse applications. For example, it is the basis of the Phred algorithm used to process trace data from sequencing machines in the Human Genome Project (https://en.wikipedia.org/wiki/Phred_base_calling). It would indeed be difficult to list all the applications for which FFTs have been used...
A periodic signal can be decomposed as a sum of several sine wave with harmonically related frequencies (frequencies at k*f0 where f0 is the fundamental frequency ).
It can be demonstrated that the Least Square estimator of the frequencies is given asymptotically (when the number of samples goes to infinity) by the location of the peaks of the Discrete Fourier Transform. Note that this technique does not use the fact that the frequencies are harmonically related...therefore, this technique is suboptimal when considering periodic signal.