Yes, you are then an online bank. This is an issue that computer games have had to figure out, when it was possible to move money both ways, cash to game gold and game gold to cash. (think Diablo's Real Money Auction House).
If you allow people to store cash value in the game, and then take the cash value back out, then you are banking. Even though it's a game, you'll have people using it as a bank, and not playing the game at all but merely using it for funds storage and movement. Even if you surcharge significantly to deter this (i.e. $1 buys 90 game gold, and 110 game gold pays $1), there will always be people willing to pay the surcharge -- criminals.
This is the whole point of the Homeland Security-driven "Know Your Customer" laws. The government doesn't want criminals using your real-money-trading platform to launder money.
I know that you imagine a business model where this would be awesome. Actually you'll spend most of your time dealing with this kind of thing:
Ann Onymous signs up and does nothing on your platform except load the account with money and send it to someone. Harold Hack is longtime customer who is active on the platform in all the normal ways. One day Harold adds a card, and loads an unusual amount of money onto the account. Both of them transfer money to Boris Badguy, who withdraws it. All fine, working as intended. But then, Ann and Harold's credit card payments reverse. They were stolen credit cards.
But you're angry at Ann and Harold, and you dun them ferociously to make good their payment per your Terms of Service. Ann is unreachable. You don't believe Harold because deadbeats lie. It takes you awhile to realize this isn't first-party credit card fraud, and Harold's account was actually hacked. Harold doesn't normally post from Moldova at 3:30am. By this time, Boris is in the wind.
Unfortunately, your business model doesn't let you make enough money to offset these kinds of losses. And it's hell on customer relations.
You finally get arrested for using customer deposits to pay payroll, because you should've been keeping it in escrow.
Being a bank is not for amateurs.