I would keep looking.
I think you’re good! I’ve met many [name_u]Rex[/name_u]'s in my life and none have ever been a dog (or any pet). And it’s a great name for a young boy, college student, mid-30’s dad, and grandpa (all [name_u]Rex[/name_u]'s that I personally know).
Besides, I’ve met more dogs named [name_u]Bailey[/name_u], [name_u]Max[/name_u] and [name_f]Zoe[/name_f] than any other name and the people I’ve met that are considering any of those names for a child have never worried about it being a dog name. Pop culture refers to dog’s as Rover and [name_u]Rex[/name_u], very few actual owners seem to. So the odds of your [name_u]Rex[/name_u] meeting a dog named [name_u]Rex[/name_u] are fairly slim as far as I can tell. And what little boy wouldn’t be thrilled to learn that they have a dinosaur name? Especially since it’s one of the cooler dinosaurs!