The concept of the 'soul' in Babylon 5 is never sufficiently explained. This is intentional; Joe Straczynski explicitly states that he did not want to resolve the issue in commentaries, special features, and interviews throughout the years, feeling it was better left for the audience to supply their own answers.
Since the concept of the soul is not sufficiently explained, it is impossible to accurately answer whether or not humans and Minbaris are actually sharing souls. This is even more problematic when you consider the triluminaries, which were given to the Minbari by the Great Machine on Epsilon 3, not created by the Minbari themselves. We do not even know if the Great Machine itself created them, or if some other species (such as the Vorlons) used Draal as an intermediary.
What we are certain of, is that Minbari and human DNA began to mix one thousand years before the events of Babylon 5, due to Jeffrey Sinclair. Delenn herself is a descendant of Sinclair/ Valen. Since Sinclair was descended from British military officers and some of the original Martian colonists, there are doubtless many humans who share enough of his DNA to set off the triluminaries, just as Delenn, a thousand years Valen's junior, did.
In short, we can't really answer the soul question. We know that some humans and Minbari do share some DNA, through Jeffrey Sinclair. We know that the Minbari believe in the concept of the soul, and that they believe the triluminaries can detect the soul of Valen, and even the souls of his descendants and relatives. Whether the Minbari are right or not, we cannot say. Joe Straczynski probably never bothered coming up with a definitive answer for that one himself.