If the list of ingredients states sodium chloride, that's what it is. Ingredients do not list reagents, just what is added to the formulation. Just about every soap product out there (shampoo, dish washing liquid, hand soap in a dispenser, body cleanser) all contain the same basic ingredients - water and one of more ionic surfactants (lauryl and laureth sulfates etc). Very few contain the sodium salts of fatty acids obtained by traditional saponification and, if they did, the saponification would be done as a separate process and the resulting fatty acid salts purified and then added to the product. Commericial products are so similar in composition that you can just as well use dish washing liquid to wash your hair and shampoo to wash your dishes (which I have). Unfortunately, few consumers bother to look at what's in the bottle.
Looking at three bottles of shampoo to hand, two list sodium chloride (at 8th and 15th positions) and one lists ammonium chloride at 4th position after water, ammonium lauryl sulfate and a betaine (lowers irritation, acts as a conditioner and thickening agent). It doesn't matter what form the chloride is as long as it dissociates to give the free chloride ion in solution. Put in the presence of dissimilar metals, corrosion is a distinct possibility.