Is this a real problem? How do banks figure out fraud from real claims? If it's a problem for bitcoin why isn't it a problem for every single thing sold on the internet?
It is a problem. Credit card fraud costs merchant/banks roughly ~$2.2B a year (2008 latest year I have a stat for).
That is a lot of cost that gets absorbed into the cost of everything we buy and pay for. That $2.2B is a combination of identity theft, plain-vanilla credit card fraud and so called "friendly fraud" (where buy decides to misreport a transaction as fraudulent/chargeback).
This is one reason credit card fees are so high. If people want the ease and access of credit cards be prepared for the massive fees and lack of anonymity that come with it.
You can't easily marry bitcoin = anonymous and irreversable with credit cards = identifiable and easily reversable. The person trading/selling bitcoins (i.e. exchange) will always lose.