The major clue is that the human-style immune system exists in all of the jawed vertebrates. But, it doesn't exist in the jawless vertebrates such as the hagfish and lamprey. And, it doesn't exist in any invertebrates such as starfish and jellyfish.
(To be more precise: we've measured some number of species, and that is the pattern we've seen so far. It is therefore a prediction of the Theory of Common Descent that we will continue to see this pattern as we measure more species.)
According to the Tree of Life project, the tree of descent from the the first vertebrate was:
============================== Hyperoartia (lampreys) | | ?== Euconodonta | | ====================== Pteraspidomorphi | | | | ?|| Thelodonti | | =====| | = ?== Anaspida | | | ====1===| | =========== Galeaspida | | | ======| | = ?== Pituriaspida ==2==| | ==3==|===== Osteostraci | ===4== Gnathostomata (jawed vertebrates)Some of the groups are known from very few (or very partial) fossils, hence the question marks. The main fossil characteristics making us choose this particular tree are:
"Dermal bone" meaning, of course, a jaw.
Unfortunately, most of these groups are extinct and it's too late to measure their immune systems. The best we can do is conclude that the human-style immune system arose somewhere along the path 1-2-3-4.
Geological dating of the fossils gives us a window about 100 million years long, with the possible dates falling in the Ordovician, Silurian and Devonian Periods. Science journalists have settled on "450 million years ago" as the simplified answer.