The mutual funds units are held by the Custodian, which is a clearly defined role as per the government regulations.
The Custodian is responsible for the safekeeping of the cash & securities in the portfolio. This role is usually played by a registered Bank such as HDFC Bank, HSBC Bank etc. Each Mutual Fund company (or AMC) has its own appointed custodian. Hence the custodian need not (and is most likely not) be the same bank as the one through which you are buying the SIP.
The following is an example for Canara Robeco Mutual Fund with clearly appointed Custodian and Registrar & Transfer Agent (latter does all paper work and some customer service).
