Matthew Broderick appears to be wearing a replica of PLA (People's Liberation Army) hat, the same one worn by Mao Tse-tung during his active days as the leader of CCP (Chinese Communist Party).
Mao Tse-tung via kwout
This historical cap, also referred to as Chairman Mao hat, was the symbol of the Red Guards. Mao Tse-tung played a major role in the establishment of the Red Army in earlier years.
Broderick also wore this military outfit in numerous scenes from the latter half of the film as Nick and the military chase Godzilla all over Manhattan and try to discover where its nest of hatchlings are.
Of interest, discover 25 Fun Facts About Godzilla, featured on MentalFloss.com.