Body wash. liquid soap, shampoo, dishwashing detergent all perform similar functions, but are they essentially the same product?

It depends on how you define essentially, but I would argue no. In fact, dishwashing detergent is often specifically used to get off oils that otherwise you'd want on your body that shampoos and body wash do not. Dishwashing liquids often contains SLS which is too aggressive for the skin and will leave you dried out. Most dish soaps and body washes also have different pH levels, and in many cases should have even more difference in their levels.

So to sum up, they're very similar in many respects, but different enough that you'd not want to use one in place of the other for long or you'd end up with stripping your essential skin and scalp oils, and potentially ruining or discoloring your dishes. Those 2 in 1 or 3 in 1 products will work okay, but for the best results you'd want to use different products.

https://dawn-dish.com/en-us/how-to/what-dawn-is-made-of-ingredients/
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4171909