BPI Family Savings Bank is a completely different bank from Bank of the Philippine Islands. They have different balance sheets and deposits made into BPI Family's accounts are not direct liabilities of the Bank of the Philippine Islands.
However, BPI owns BPI Family Savings Bank. In terms of business focus, BPI Family Savings is run as a "consumer banK" catering to "smaller" customers.
From my experience, BPI Family tends to pay more for deposits than the comparable products at BPI, and charges more for loans than the comparable products at BPI. The minimum balance requirements are also lower at BPI Family.