For plants and pests, a diluted mix can help deal with soft-bodied insects like aphids. But this needs to be used carefully—too much soap can damage leaves. It’s more of a controlled solution than a casual spray.
Some claims need caution. Using dish soap on pets, for example, is only okay in specific situations (like emergency flea removal), and not as a regular shampoo—it can dry out their skin. The same goes for hair: it can strip color, but it will also strip natural oils and leave hair rough.
As for things like poison ivy oils, dish soap can help wash them off skin or objects if used quickly, since it breaks down the oil. That part is actually backed by how the soap works chemically.