Write a formula to obtain a similar result to a pivot table summary

You could use a sumproduct.
You first just need to make your row labels. I honestly just copied and pasted your pivot table values and used those.

=SUMPRODUCT(((TEXT($K$2:$K$27,"mmm")=$F32)*(TEXT($K$2:$K$27,"yyyy")=$F$31))*$J$2:$J$27)

"mmm" above is used if only looking for the month in 3 character acronym. change according to your row label set up.

F31 above can be chamged to just "2020" or "2019" You can also just chamge the cell reference if you want that somewhere else.

Actually I should have done a better job of displaying the formatting I would like to use. I did not want to use a pivot table at all. What I would like to do is just use Columns X and Y (or any others) and write a formula in column Y based on the dates in column K and the quantities in column J. Hopefully this makes sense. I was not sure how to adapt what you provided into this. I am attaching an updated spreadsheet - note I have hidden some columns for neatness. Thank you

Pl see file. x2:X13 formatted for mmm.
In Y2 then copied across

=IFERROR(1/(1/SUMPRODUCT((YEAR($K$2:$K$27)=Y$1)*(MONTH($K$2:$K$27)=MONTH($X2))*($J$2:$J$27))),"")

Pl see file. x15:X48 formatted for Mmm-yyyy.
In Y15 then copied down.

=IFERROR(1/(1/SUMPRODUCT((YEAR($K$2:$K$27)=YEAR($X15))*(MONTH($K$2:$K$27)=MONTH($X15))*($J$2:$J$27))),"")

Thanks for feedback and rep.