Hi Chris,
The LOOKUP functions (including MATCH) will return only the result associated with the first occurrence of the search value they are looking for.
Your formula: VLOOKUP(A$1,September Hours::$A4:$E4,2,FALSE)
expressed in English: Get the value in A1 of 'this table',
Search for it in the first column of the cells A4 to E4 of September Hours,
If found, return the value from the second column of A4 to E4,
Accept only an exact match.
Your formula defines a lookup table of only four cells. The search is confined to cell A4 of September Hours. The return value can come only from B4 of that table. If an exact match of the search-for value is found in A4, then the formula will return the value from B4. If not, the formula will return a 'can't find it' error message.
When filled down, the formula will continue to get the search for value from cell A1 of 'this table', but will again search only one cell of column A of September Hours for a match, and return only the value in the same row of column B.
Assuming the bottom table is the one containing the formula, and the top table in your post is September Hours, the formula you present must be as it appears in B4 of the bottom table, where it returns "Katy", the name in B4 of the upper table.
VLOOKUP (and the other lookup functions) can search for only a single value, so asking it to find the customers by Cleaner AND Preferred Day requires combining those two values into a single search-for value.
VLOOKUP (and the other lookup functions) will return values from only the row in which they first find a match for the search-for value.
To find multiple occurrences of the same value, as in his case, requires producing a list of 'index values' marking each occurrence of each value with a distinct value.
VLOOKUP is restricted to using the first (leftmost) column of the lookup table as its search-where location. So using VLOOKUP means the matches for the combined search-for values must be in the first column of September Hours, if VLOOKUP is to be used.
Other functions in the lookup family do not share this limitation, and my own choice would be to use the MATCH and INDEX pairing for that reason.
Here is a copy of your September Hours table (renamed 'Data' only to shorten the display of the formula).

The formula shown is entered in E2, then filled dows to the end of that column.
The index list it creates is a set of text strings combining the cleaner name, the preferred day and the count of that combination to 'this row' in the table.
The second table, Summary, collects a summary list of the customers of a single cleaner (named in A1) for each day.
The Test version of this table contains no error trap, so it displays the expected error messages in cells where there is no match for the search-for value in the index column above.

The formula is entered in B2, then filled down to the end of column B and right to column D.
After determining that no unexpected errors were present, an error trap was added to the formula to trap the error messages and place a null string in the cells containing those errors.

The new formula is filled right to column C and down to the last row of the table, replacing the previous version.
Column E of the Data table should be hidden. It's not needed by viewers of the table, and hiding it will help protect the formula it contains.
Regards,
Barry