Yes. That uniquely identifies your account. The bank's name is not required because anyone can work that out from the sort code. For example, my current account has a sort code that begins with 30 and that is immediately identifiable as Lloyds Bank. You could easily check that - lists of British banking sort codes are online. What someone paying you MIGHT want to know is the name on the account, but that would only be used as an extra check to make sure nobody copied the number wrong! Otherwise you'd only know about it when you don't get paid and someone else suddenly finds they've got your wages.
Just out of interest, sort codes were invented to identify bank branches in the days before computers and networks. So each branch had its own list of account numbers and could immediately open an account for you with its own number without having to check with head office. Every account at the same branch would have the same sort code. So you'd not only know which bank it was from the sort code, but which branch in which town.
This also helped with physical clearing of cheques. They all come together at the Clearing House, are sorted into piles with the same sort code, then each pile can go to the right branch to be cleared (the branch decides whether it can pay this cheque, or if that would put the account too far into overdraft, bounce it back). See why it's called a sort code?
All very different now - I don't even know whether clearing happens this way any more, and with computers and network communications, it probably doesn't - so especially with newer banks, the sort code might not have any direct relation to any branch. If we were to invent a new banking system from scratch, we probably wouldn't do it this way. (Reminds of the old Irish joke - an Irishman is asked directions and he answers "well, I wouldn't start from here".) But we're stuck with it now and the full sort code always DOES identify the bank.