According to this source, the game of "Twenty-One" didn't appear in American gambling halls until 1919. The origin of the name, Blackjack, is explained as follows:

So how did "Twenty-One" become "Blackjack"? Apparently when the game was first introduced in American it wasn't very popular so the gambling houses tried various bonus payouts to get the players to the tables. On such bonus was a 10-to-1 payout if the player's hand consisted of the Ace of Spades and a black Jack (either the Jack of Clubs of the Jack of Spades). This hand was called a "blackjack" for obvious reasons and the name stuck even though the bonus payout was soon abolished.

In view of this and the age of the song, it seems unlikely that the name used was in reference to the card game.