(disclaimer: not an aeronautic engineer) when you double an object, its weight increases by 8x (all three dimensions increase by 2x) but the wing area surface only becomes 4x larger. You thus end up with a worse lift-to-weight ratio.
In addition, the purpose of a regular plane is to transport goods and people while the purpose of a paper plane is to just float. The closest full scale objects to a paper plane would be gliders, which do ressemble paper-planes to some extent.
