SNR can be used/provided for a variety of different reasons. However, for the characterization of a signals property, the signal to noise ratio may or may not be important as you can always do an estimated removal of background noise from the signal before characterizing its properties. However, this removal of background noise will have a minimal effect on the signal properties unless the SNR is quite low, i.e. approaching masking of the signal, as generally the subtraction of background noise in the units of uPa^2 is often negligible in these circumstances. There is no rule of thumb of an appropriate SNR that I am aware of, but you would want the signal to be well above background noise across its frequency composition to ensure that things like PSD reflect the signal and not the noise. Perhaps 6dB or higher would be appropriate. So while that is important, it is also important that the characterization of a signals is based upon signals that have not been subjected to high transmission loss across its spectral components. For example, if you were characterizing a signal, you would ideally like that its frequency composition be present so you can get a good picture of its actual PSD, etc. So I would say that SNR and signal amplitude are both important to ensure the results portray the actual signal vs. noise and impacts from varying levels of transmission loss as a function of the signal's frequency composition.
Lastly, for detection and presence data, it is often useful to provide an estimate of the SNR required for a signal to be detected. This is true for both automated detectors and manual review. For manual review of spectrograms, etc., different people require different SNRs of signals to "see" them amongst background noise. As background noise changes often as a function of time, if SNRs of detection are static then this change in noise levels effectively changes the sampling range of a sensor. It can get complicated, but it is something to consider when analyzing results.