Olaf tries a paddle ball for a nose. The paddle ball wasn't invented until the 1920s.
Oaken shows Olaf what appear to be color photographs. Color photography wasn't invented until 1861, and 'Frozen' takes place in 1843.
Olaf tries a snow globe for a nose. The snow globe wasn't invented until 1900.
One of the pictures Oaken shows Olaf is of a beach scene with a beach ball shown. The beach ball wasn't invented until 1938. The picture also shows a plastic bucket and spade; plastic wasn't invented until 1907.