I have come across cryptocurrency exchange sites (for example, coinbase, bittrex, etc) and I was wondering if they operate like banks. What are the (usual) legal requirements in order for a site to operate legally?
Main laws that they do have to comply with are Money Laundering and Terrorist Financing and "Know Your Customer" type rules.
This is why when you open an account you have to send ID etc.. These they are forced to comply with as the banks that serve them (remember you have to wire them the USD EUR etc.. somehow - and that is through a bank with whom they have an account) force them.
What they DO NOT have to comply with are all the other regulations that an exchange/bank would normally have to comply with - hence the reason they go bust so often!
E.g. they are not required to segregate client funds, they do not have to retain capital to protect clients, nor do they have the same requirements for auditors, segregation of duties (making sure the developer is not supporting the site too!), being fair to clients etc..
Each site and it's country regulation rules. if there is no regulation or law, for the majority there isn't, so the site can be forced to be closed someday.