In both games, the winning strategy is to move in such a way that the arrangement of the dominoes on the board is symmetrical at the end of your move. This guarantees that if the other player can find a place to play a domino, you can do so as well, simply by playing in the symmetrically opposite location.
In game 1, where dominoes must be aligned with squares, the first player can't do this. The first domino placed on the board must be off-center, and the second player can then play symmetrically opposite to restore the symmetry. The players alternate moves until eventually the second player fills in the last place where a domino can be played, and wins.
In game 2, the first player can place the first domino exactly in the middle of the board, and the second player's move destroys the symmetry. Now the first player plays opposite the second player, and eventually wins after filling in the last available place.