Paul is sent to Forest Manor School, which is overseen and financed by Frederick Trump. Trump never oversaw or financed a school. He sat on the board of Kew-Forest, which his children Maryanne and Donald attended. That said, this *goof* was likely intentional on the part of the filmmakers.
They are wearing long sleeves and long pants on the first day of school. The weather in Queens calls for shorts and short sleeved shirts. No one would wear a turtleneck and a flannel like Paul is in early September.