Yes, the US does have a similar concept. It’s called the routing number.

To tell a fun anecdote: My parents have two bank accounts with identical account numbers and different routing numbers. They still use both accounts for legacy features of each.

The accounts used to be a single account at one financial institution. It got bought by a larger financial institution about ten years after they were married (this was their first joint account as a couple). Fifteen years after that, the bank split apart into three regional banks. The routing number changed with each corporate change but the account number persisted. In the last split, some customers who had property mortgages were “retained” by the divesting bank and “acquired” by the divested one.

Thus, two identical account numbers, different routing numbers.

A sort code is a six digit code that identifies the bank and the branch of that bank where the account is held.

It's called a bsb in Australia. First six digits