Your suspicions are correct . . . They are wrong.
Their answer would be correct if the game were like this:
The two players select a card at the random from their respective decks.
Then the cards are returned to their decks, the decks are shuffled,
. . and they draw another random card.
And they repeat this procedure fifty-two times.
The game, as stated, involves the number of derangements,
. . arrangments in which no object is in its designated position.
This number has a complicated formula and a lengthy derivation.