Symbolism, Imagery, Allegory
One of the first things Daniel notices about Natasha is that she's wearing a leather jacket with Deus Ex Machina on the back. He remembers that the phrase means "God from the machine."
Here's a Nicola Yoon-esque backstory a la Shmoop: The term deus ex machina originated in ancient Greece, when playwrights used simple machines like cranes or lifts to whoosh in actors playing gods to resolve the conflict at the end of the show.
For a more modern literary example, (spoiler alert) Catching Fire from the Hunger Games trilogy has another example of a deus ex machina twist: Katniss is about to die in an fiery explosion when she's whisked away by a hovercraft. In the real world, it would be like forgetting to study for a major test, only to have school called off due to an unfortunate incident involving a creative chemistry professor and a group of angry bumblebees.
You get the idea: Conflict arises, things look hopeless, the gods appear, and everything is magically better.
Some people criticize the use of deus ex machina as a plot device because it feels too contrived and unsatisfying, like the author just couldn't come up with a more creative ending. However, the playwright Euripides was the president of the Deus Ex Machina Fan Club, using the technique to end many of his most famous works. Surely we can't blame all of those endings on a lack of inspiration, so it's possible that Euripides knew his Greek audience would understand that when a story was set in motion by the gods, it only made sense for the gods to end the story as well.
So what's symbolic about Natasha wearing a Deus Ex Machina jacket? A lot.
First of all, Natasha herself would probably be the least likely character to believe in deus ex machina endings, given that our resident cynic claims not to believe in God or fate or events happening for any sort of reason at all. She never acknowledges the message on the back of her jacket, so she may or may not have been trying to make an "ironic angsty teenager" statement. However, Daniel notices it right away, and his poetic brain can't help but wonder if it's a Sign-with-a-capital-S, since the evangelical conductor just told him to go look for God.
Natasha and Daniel's magical day in NYC almost has a deus ex machina ending of its own: Attorney Jeremy Fitzgerald should have been able to get Natasha's deportation order delayed at the very last minute, but he was too busy getting busy with his paralegal to actually make it to court.
Of course, there's the added twist that Natasha and Daniel's story does, in fact, end up with a much more convenient happy ending. Ten years later, the long-lost-star-crossed lovers just happen to be sitting only one row apart from each other on a plane. Their flight attendant just happens to be Irene, the former security guard who decided not to kill herself after a random thank you from Natasha—a decision that could be considered another deus ex machina moment all on its own. Irene tells Natasha how their stories are connected, Daniel overhears, and the protagonists are reunited once again. Maybe "the machine" was a Boeing 747 all along.