Answer
2/3
Reasoning
In randomly picking a card and randomly picking a side from that card, you have effectively picked a random card-side from all the card-sides in the hat. There are 3 white card-sides in total. 2 of which have another white side on the back. Given that you've picked a white card-side, it follows that there's a 2/3 chance you've picked one of the white card-sides with another white side on the back.
This reasoning extends to any number of cards with any number of colours:
Given that a randomly picked side of a randomly picked card is colour A, what is the probability that the opposite side is colour B?
Count the total number of card-sides of colour A. Call this $N_A$. Then count the number of those card-sides with colour B on the back. Call this $N_{A,B}$. Then the probability of the opposite side being colour B is $N_{A,B}/N_A$.