Banks are basically involved in the function of savings and loans and on the other hand Mutual Funds perform the function of investment. When you keep your money in a bank savings account or FD, you are saving whereas when you put your money in mutual funds, you are investing. Banking and Mutual Funds are two completely different business and functional areas with different organizational characteristics. Banks are governed by RBI whereas Mutual Funds are governed by SEBI. If a corporate (corporation) wants to come into the field of banking and mutual funds, it will need two separate license permits from the relevant regulators, and will be able to operate and do business as two separate companies.
You must have also grown up with such banks which also do the work of mutual funds. Both of them have completely different scope of work, both are separate companies with no mutual functional relationship. Bank's own good performance does not mean that the returns of mutual funds of the same name will also be good. Both are completely unrelated fields.
Today, most of the banks distribute various financial products among which mutual funds are one. Banks act as a selling channel for mutual funds that have distribution tie-ups with them. So, when you want to approach the bank for investing in mutual funds, keep in mind that the bank does not have all the investment options in mutual funds that are available to you in the market.