Two separate components for two different granularities — pick a month, or pick a day. Neither has a built-in dropdown; both are typically wrapped in a Popover with a button trigger.
Reach for Month Picker when the input only needs a month and year — a reporting period, a billing cycle, a traffic-projection window. Reach for Calendar for anything that needs an actual day — a booking date, a due date, a date range. Neither component animates its own open/close; that comes from whatever Popover wraps it.