Could 'A Midsummer Night's Dream' be classified as Greek mythology?

No. There are characters from Greek mythology, in the persons of Theseus and Hippolyta. Titania is another character from Greek myth.

However the origins of some of the other characters is not Greek myth. In English folklore, Puck is a mythological fairy or mischievous nature Sprite. Puck is also a generalised personification of land spirits. In more recent times, the figure of Robin Goodfellow is identified as a puck. Oberon is a character from French folklore.

In terms of sources, A Midsummer Night's Dream is a bit of a dog's breakfast.