I think that XRF can be used to determine the elemental composition of steel and provide a fairly accurate "fingerprint" of the steel sample. It is a surface analyses technique and the sample tested needs to be big enough. These days they have hand-held analyzers that can provide almost "instant" results. Sulfur and phosphorous may not be detectable that way.
If only a small sample is available, it could be dissolved in a mineral acid and analyzed using ICP-AES which I think can detect and measure all of the elements Edgar listed. I am not a metallurgist but I would venture a guess that the elemental composition would be the same throughout the test sample except, perhaps, for the carbon content. Again I am guessing but I suspect that the carbon content may vary widely within the sample, i.e. higher at the surface compared to at the center of the test sample.
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