Account number are Unique based on the following combinations, based on manual banking practise of early days;
-Country -Bank -Branch
-Location
-Account
Number
-Currency
-Account Type [Savings, Checking]
With the current computerization, there are quite a few items that have been coded into the number, for example Branch, Currency, Account Number and Account Type are part of account number.
Overall there are some attempts to standardize the account number so that they can be unique across the world as described as littleadv.
money.stackexchange.com/users/1113/Dheer
Within the bank or group of banks owned by the same company the account number is unique.
What is unique is the bank id number and account number combination.
If the bank ID number doesn't match a banking institution, the transaction will not be completed.
If you are unlucky and your mistake in writing the bank id number does match a bank, you have to hope the account number doesn't match.
The name's do not have to match.
Checks have been deposited with incorrect spelling, nick names vs formal names.
Mr. and Mrs ...
money.stackexchange.com/users/5414/mhoran_psprep
There are two unique identifiers for a bank account: SWIFT code + bank internal identifier, or IBAN code.
IBAN is mostly used within European banking system, and the whole code provides a direct and unique identification of the account.
SWIFT is an international network where each bank/bank branch has its own address, and account number is a metadata added to the message for the receiving bank to handle.
Usually the name of the recipient and additional information are required when wiring money through the SWIFT network, to match the records and make sure there's no mistake.
Account numbers don't have to be unique, not even within the same bank.
There's always something else in addition to uniquely identify them.
money.stackexchange.com/users/2998/littleadv
This is just my experience
but I think I have to put in the account holder's name or last name to check that I'm sending it to who I think I'm sending to.
money.stackexchange.com/users/6061/theproxy
From personal experience, I can tell you that bank account numbers are not unique.
Someone from another branch of my bank was able to withdraw money from my account at my branch because they had the same account number.
You are supposed to enter your branch number on the withdrawal slip in front of your account number.
The person who got my money did not do this.
Because it was at my branch, the teller debited my account for the transaction.
I caught this on my monthly statement and immediately complained to my branch manager.
He was able to retrieve the withdrawal slip and saw what had happened.
He credited my account and said he was going to talk to the teller who should have asked for the branch number and/or should have noticed that the name and address on the withdrawal slip did not match those on my account.
I would not have thought that the bank would allow this situation considering how many numbers are available to assign but they did.
money.stackexchange.com/users/6755/Barry Schwartz
If somebody sends me money and I write the swift code wrong, will the money go somewhere else?  I know bank account numbers are unique within a bank, but will are they unique globally?  Update: I found a case titled  Money withdrawn to wrong bank account and no response from Paypal  (fortunately not my case)
Are bank account numbers unique?
money.stackexchange.com
<banking><bank-account><deposits><world>
money.stackexchange.com/questions/15528
money.stackexchange.com/users/6296/user4951
The bank id number and account number combination will be unique.  Most people believe that within a bank or group of banks owned by the same company that account numbers are uunique, but some people have had the unfortunate experience that this is not the case. What is unique is the bank id number and account number combination.  There are two unique identifiers for a bank account: SWIFT code + IBAN code.  
Most people believe that within a bank or group of banks owned by the same company that account numbers are uunique, but some people have had the unfortunate experience that this is not the case. What is unique is the bank id number and account number combination.  There are two unique identifiers for a bank account: SWIFT code + IBAN code.  