A Pivot Table would do this for you very quickly...
Set your Date Field as both Row Label and Date Field (set to Count)
Right click on a Date in the Row Label and choose "Group" select "Month" only.
Formula wise using the values as you have them you would need a SUMPRODUCT formula...
Assuming only dates/blanks exist in the range reviewed then, eg:
If it's possible non-numeric/non-blank value exist in the range then you would use:
Note were this the case then the Pivot route would not work either
(you would not be able to Group the field by Month)
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