Perhaps he did not have a passport. Were his family travellers? Where I live, children do not have their own passports until they are 16. Their details are added to a parent's passport before that.
Its very possible he doesnt. He would have to have his own if he traveled out of the country, though, you arent just added on to a parents passport here.
I think it was said that his dads recent vacation was out of the country. I wonder if Logan had traveled internationally with dad before.
Additionally, if he really wanted to get out of the country, he could have gotten across the border into Mexico much easier, no? ID-wise? I think anyone can just walk into Mexico from the US at the border-crossings, right? It would have only been a few extra hours of drive time.