Two way sum lookup (by month & name)

Try in cell B12:

=SUM(FILTER(FILTER($B$4:$W$7,(MONTH($B$3:$W$3)=MONTH(B$11))*(YEAR($B$3:$W$3)=YEAR(B$11))),$A$4:$A$7=$A12))

and copy down and across.

This task becomes a lot easier when the data is flattened out. In this example we start with "pivoted" data.

Two different ways to do this:

1. Load data into PowerQuery, unpivot to flatten out (and set date column to be treated as dates), and then create pivot table.

Unpivot columns - Power Query | Microsoft Learn

2.  UnPivot with a Lambda and then use PIVOTBY to aggregate the data.

=LET(
    flat, UnPivotMλ(names, weeks, data),
    name_col, TAKE(flat, , 1),
    dates, CHOOSECOLS(flat, 2),
    month_year, DATE(YEAR(dates), MONTH(dates), 1),
    val, TAKE(flat, , -1),
    PIVOTBY(name_col, month_year, val, SUM, , 0, , 0)
)

Where UnPivotMλ is a generalized Lambda which unpivots a matrix by providing rows, columns, and values:

UnPivotMλ = LAMBDA(rows, columns, values,
        LET(
            i, ROWS(rows),
            j, COLUMNS(rows),
            c, COLUMNS(columns),
            s, TOCOL(SEQUENCE(, j) * SEQUENCE(c, , 1, 0)),
            row_labels, WRAPROWS(TOCOL(CHOOSECOLS(rows, s)), j),
            attribute, TOCOL(CHOOSEROWS(columns, SEQUENCE(i, , 1, 0))),
            v, TOCOL(values),
            HSTACK(row_labels, attribute, v)
        )
    );

I know some will say this is too much but if you encounter data that needs to be fixed before analyzing, I think it makes sense to have a function in your library on hand to flatten out data. It takes a lot of the leg work. No need to repeat common cleanup tasks.