Sum if month - year and name is the same

hi unzmannn, welcome to the forum. try this in B8 & copy to C8:
=SUMPRODUCT(((TEXT($A$2:$A$5,"mmmm")=$A$7)+(TEXT($A$2:$A$5,"mmmm")=$A$8))*($B$2:$B$5=B$7)*($C$2:$C$5))

note that this ignores the year since A7 & A8 do not have it. also, both A7 & A8 must be months spelled in full.

Create a pivot table and group the dates by year and month. No formula required, just a few clicks. See attached.