All the ranges need to be the same size with COUNTIFS so to use that function you need to use it twice, once for each column, i.e. this formula in I5 copied down

=COUNTIFS(B:B,H5,A:A,">="&I$2,A:A,"<= "&I$3)+COUNTIFS(C:C,H5,A:A,">="&I$2,A:A,"<= "&I$3)

....or if you prefer to avoid the repetition you can use an "array formula" for the same result, i.e.

=SUM(IF(B$2:C$300=H5,IF(A$2:A$300>=I$2,IF(A$2:A$300<=I$3,1))))

confirmed with CTRL+SHIFT+ENTER