With sophisticated phishing attacks that are now hosted on HTTPs servers, this practice is just lunacy and it happens in every bank. Big small, private, public all engage in this nonsensical practice. Among the biggest names: SBI, YesBank, Central Bank are the ones that do this. I know this because I have inquired how to open accounts at these places.
I actually have a dedicated personal relationship manager who filled out all the forms! And even now for certain operations like booking currency forwards(FCNRs) they ask you to fill forms and send it over the email. Not to mention its inefficient as well.
Its a testament to two things: 1. There are so many people in India that banks just find it cheaper to hire a person and let him interface with you instead of designing a web application. 2. No one gives a shit about privacy. 3. The system is wide open for people to Man in the Middle and Phish.