Saying that bitcoin allows users to act as their own banks is like saying a mattress with your cash under it is your own bank. It's not meaningful in anyway. Debt is the meaningful differentiator.
I can buy $1 worth of Bitcoin, declare it ErikCoin, acceptable only in my stores at a 1/1,000,000 denomination, and I instantly "minted" $1,000,000 in cryptocurrency, using only the Bitcoin blockchain.
Bitcoin doesn't let people be banks, it let's people be the mint.
- You can't destroy anyone's balance after you've issued them currency
- Balances are publicly verifiable
- Balances can be stored in one's mind
... lots of things. Tons of affordances differ from a game currency.