Music fingerprinting algorithms (like the one used by Shazam) will not work. They work by identifying areas of the spectrogram in which there are sharp increase of energy in a narrow frequency band (the attack of a note partial) - and "noisy" sounds like breathing never have such features.
Cross-correlation will not work either because of the predominantly noisy component in breathing signals. For example, two realizations of white gaussian noise filtered by the same band-pass filter will sound absolutely identical but the correlation between waveforms will be null. The notion of similarity given by our ears, or by the experience of a physician, is not waveform correlation.
It is not clear from your question if what you want to measure is just how "clean" the signal ("is it similar enough to this clean recording of breathing?") or if you will actually need to match a recorded sample to several reference recordings (for example for diagnosis). In both cases, you will have to define your own features using your own expertise of the field, and probably rely on statistical/machine learning techniques.
The former situation is easier because temporal information probably doesn't matter much. It seems doable with features like spectral envelope (for example, energy in a handful of frequency band), a contrast measure on the spectral envelope (smooth the spectrum, compute kurtosis or geometric/arithmetic mean ratio), and a contrast measure on the correlation (to discriminate against sounds which are too "pitched" - just like voiced/unvoiced detection in speech analysis). From there, you could have a few examples of "good" and "bad" signals, and train a classifier.
For diagnosis, you would have to define features taking into account temporal information (such as the autocorrelation of the envelope).