Either A) New Years Day is a day of the week and your birthday changes every year (but in a vastly more predictable way; NYD will make next year’s dates one weekday ahead), or
B) New Years Day is a completely separate day and all years are identical, and you choose your birthday to be celebrated on the closest agreeable weekend if that matters to you
People with their birthday on Monday are not going like that their birthday is always on Monday…
Either A) New Years Day is a day of the week and your birthday changes every year (but in a vastly more predictable way; NYD will make next year’s dates one weekday ahead), or B) New Years Day is a completely separate day and all years are identical, and you choose your birthday to be celebrated on the closest agreeable weekend if that matters to you