The isblank function is intended for use with an individual cell; its behavior when applied to a range is not documented as far as I know. In your case, isblank($B:$B) returns True if it's placed in a row where B column is empty.

Use counta(...)>0 instead, like this:

=IF(COUNTA(Sheet1!$B:$B)*COUNTA(Sheet1!$C:$C)=0, 0, SUMPRODUCT(Sheet1!$A:$A=F2, Sheet1!$B:$B=Sheet1!$C:$C))

Short answer

Alternative 1

Instead of {"First Value;"Second Value"} write ={"First Value,"Second Value"} in cells D2 and D3 in the example provided in the question.

Then, instead of using a category names reference of the form D2 use the D2:D3 reference form.

Alternative 2

Instead of {"First Value;"Second Value"} write First Value;Second Value in cells D2 and D3 in the example provided in the question.

Then, instead of D2 as the reference in the formula use =SPLIT(D2,";")

Explanation

1. When an array in wrote in a cell without a leading formula operator (= or +) it is interpreted by Google Sheets as a string instead as an array.

Alternative 1

2. As the OP want to put one array in each row, instead of using a row separator, use a column separator.

3. As the array values are in a 1x2 range the reference to use in the formula should be of the form D2:E2. The formula will be like the following

   =ArrayFormula(sum(sumif(A:A, D2:E2, B:B)))

Example

Alternative 2

In order to make easier to convert a string to an array it's suggested to remove the enclosing braces and the quotes, otherwise more functions will be required to remove them and will make the spreadsheet more complex and slow.

The formula will be like the following:

=ArrayFormula(sum(sumif(A:A, SPLIT(D2,";"), B:B)))

Example