When you "have an account" with a bank or a company, it means that there has been a customer relationship set up between you and the company. You and the company are able to pay money back and forth through this "account". Your bank account is one example. When you put money into your bank account, the bank keeps a record of how much money it owes you.
We use the word "open" to talk about starting such an account:
I'd like to open an account, please.