When you play with the house's money, you are putting your original investment back into your pocket where you have no risk of losing it and are keeping the proceeds from your investment in some security (investment vehicle), where even if you lose it all, you still have your original investment amount.
The term actually applies more to gambling than it does investing. The "house" is the casino. And, let's say you go to a blackjack table and start playing with $100. You may lose some or you may win some. But, if you win, say $50, and you now have $150, you could put your original $100 back in your pocket and just play with the $50 you already won from the "house" (you're "playing with the house's money").
With investing, using your scenario of buying shares that go from $1 to $2 a share, you have to pay taxes on all gains. So, if you sold 50 shares at $2 a share, you had a "gain" of $1 for each of those shares sold. So, you would owe capital gains taxes on $50 of the $100 in proceeds you got from selling 50 shares at $2 a share.
So, you should be able to see that "playing with the house's money" doesn't really apply with investing quite like it does with gambling. But, the concept is the same. You invest an original amount, your investment increases in value, and you take out the gains that equal your original investment, plus any taxes you have to pay on those realized (cashed in) gains. Then you just let the money your investment made for you be the money you keep in the investment, and you then have nothing to lose.