Based on @Michal answer, the formulae to use for the example are:

• Number of families that have their eldest child in school A:

=SUMPRODUCT((C2:C9=MINIFS(C2:C9,A2:A9,A2:A9))*(D2:D9="A"))

• Number of families that have their eldest child in school B:

=SUMPRODUCT((C2:C9=MINIFS(C2:C9,A2:A9,A2:A9))*(D2:D9="B"))