Do post offices also open a Demat account along with a savings account in India?

You cannot open a demat account with a post office at this time.

You can invest in mutual funds though, as the Post Office Savings Bank has tied up with UTI Mutual Fund to distribute its products.

However please beware that banks in general offer only regular plans. So you will incur significant commissions that erode your wealth in the long run (1% to 1.5%). You are better off with a financial planning platform that offers direct mutual funds.