Copy and paste multiple columns with specific date to another worksheet

I think this macro will do what you want...

Sub DistributeMeterRecordings()
  Dim R As Long, MnthName As String
  MnthName = Format(Cells(R, "A"), "mmmm")
  For R = 1 To Cells(Rows.Count, "A").End(xlUp).Row Step 96
    Intersect(Sheets(MnthName).Rows(R).Resize(96), Columns("A:G")).Copy Sheets(Format(Cells(R, "A"), "mm-dd-yyyy")).Range("A1")
  Next
End Sub