Imagine there are only two cards, card a and card b. If you deal them the way OP's solution did, ie one at a time, there are two scenarios: card a is first and card b is second, or card b is first and card a is second. This equates to (2c1)(1c1)=2 ways to draw the cards.

But if we change the strategy to be like in poker, ie draw 2 at a time, there is now only one way to draw the cards, drawing a and b. The order doesn't matter, since a hand of b and a is the same hand in the game. This is the (2c2)=1 case. If you notice, there are half as many ways when we use the second strategy. The reason for this is getting rid the importance of order. If we cared about order, both cases would be the same, but since order doesn't matter for the first draw in poker, we must disregard it here.

Now just switch it to starting with 13 cards, and you'll still find that (13c2)=136 which is half as many as (13c1)(12c1)=1312 if we draw back to back