Just like the Oracle said, Dilan didn't really trust her words. The Oracle knew way too little information about the Fate Changers to be able to conclude that he was in possession of the same True Undying Essence as someone she knew.
It was too much of a coincidence, even more so when she told him a bunch of additional pieces of information.
The Oracle never liked him, to begin with. Thus, there was no reason for her to reveal several secrets, all of a sudden.
'Who cares if some Void Walker, Titan, or whatever he may or may not be, has the same True Undying Essence as I have? Does that concern me? Not really!'
Dilan was not sure if he could trust everything the Oracle said, but it was also a fact that she gave her utmost to make sure that her Universe would survive. With that in mind, he could tell that there were a few things he could trust.
However, in the end, he would have to figure out a few things by himself.