No what they are doing is hashing each character and storing that or storing the password in clear in hopefully a secure device e.g. HSM.
It is not a bad approach; the bank requests the user enter a number of characters (e.g. HSBC is 3 from random positions). It does mean that if a trojan, key logger or phishing site captured those 3 characters they do not have the full password.
It also allows the bank to use the partial password for authenticating users over the phone etc without relying on secret questions or a different password, again without the call center person knowing the full password
A few problems which is probably why it is not more widely used:
The main one is that you do not want to store the password in the clear (in fact regulations like PCI-DSS forbid it). So the approach is to create a one way hash of the password and store that. When the user enters the password, this is hashed and compared to the stored hash, if they match the user is authenticated. With a partial password you either have to store the password with reversible encryption which is not best practice or store a large number of hashes e.g. for HSBC with a birthday plus every character of the password requires a separate hash. You have to also impose a maximum password length so that you can allocate the fields in the database to store all the hashes (although there are probably smarter database and application technologies now that would get around this)
It encourages users to select weak passwords e.g. if I have a 8 character complex password: tpEz%e2S . Trying to remember what the 2nd, 4th and 5th character of that is difficult, which encourages users to select a simple dictionary word.
Also if there is no account lockout type features it is exponentially easier to guess or brute force 3 characters than 8 characters.
Confidence that the full password is not known could be misplaced e.g. if the password is Fulham and you know F, l, h, you maybe able to make a decent guess at the full password
A phishing site that simply asked for the 1st, 3rd, 5th then said password incorrect and changed to ask for 2nd, 4th, 6th could also get the full password as user would probably enter it before they thought twice
From a user convenience perspective it means they cannot use the browsers remember password or a password manager like Lastpass or 1Password
Most banks are moving away from relying on a username and password e.g. HSBC offers an RSA token for business customers, Barclays have EMV smart card reader, almost all of them use detection technology like RSA adaptive authentication to establish a baseline of browser, location, usage profile, velocity etc for passive authentication and unauthorized use detection.