What is the scope of a mechanical engineer in the banking sector, specifically in public sector banks in India?

To be terse, Banks dont care about what you study per se for generalist role. Exceptions are there if they hire you as consultants and not as employees if you are very renowned in any particular field. I am assuming question is for a fresh engineer. Banks are in the business of building and maintaining trust with clients, customers. Since banks especially private ones are filled with engineers. People trust familiar backgrounds to an unfamiliar background. Hence there might be more engineers than non engineers in a group of ten, lets say amongst bankers. Hence you might have an advantage. But this is contingent on you being better at problem solving and idea generation than lets say, a non engineer. These skill sets are open to anyone as nobody teaches these specific things in any course. Some courses make you more familiar to these skill sets than others. Thats as far as the advantage goes.

Probation period is 2 years in most of the banks. You have to clear some written tests within those 2 year before you are absorbed in that bank. The job is hectic, pay is good, better than private sector jobs. Kindly google if you want exact package. A banker also gets a huge respect in the society as he is regularly consulted about the different mutual funds, investments, etc. I can vouch for the 'respect in society' part. For bank PO you just need to be a graduate. I am a mechanical engineering student, my dad is an AGM in SBI.

Most banks these days are all based on networking, hardware etc.  If you're into a turnkey (mechanical engineering) you would try something like integration of mechanical engineering and software (Embedded Systems) Most banks employ Embedded Engineers who are very useful for their online, phone, e-banking sectors.