First we'll perform a small refactoring and extract the seat designator parsing and validation logic into it's own method, _parse_seat(). We use a leading underscore here because this method is an implementation detail:

class Flight:

    # ...

    def _parse_seat(self, seat):
        """Parse a seat designator into a valid row and letter.

        Args:
            seat: A seat designator such as 12F

        Returns:
            A tuple containing an integer,string for row and seat.
        """
        row_numbers, seat_letters = self._aircraft.seating_plan()

        letter = seat[-1]
        if letter not in seat_letters:
            raise ValueError("Invalid seat letter {}".format(letter))

        row_text = seat[:-1]
        try:
            row = int(row_text)
        except ValueError:
            raise ValueError("Invalid seat row {}".format(row_text))

        if row not in row_numbers:
            raise ValueError("Invalid row number {}".format(row))

        return row, letter

The new _parse_seat() method returns a tuple with an integer row number and a seat letter string. This has made allocate_seat() much simpler:

def allocate_seat(self, seat, passenger):
    """Allocate a seat to a passenger.

    Args:
        seat: A seat designator such as '12C' or '21F'.
        passenger: The passenger name.

    Raises:
        ValueError: If the seat is unavailable.
    """
    row, letter = self._parse_seat(seat)

    if self._seating[row][letter] is not None:
        raise ValueError("Seat {} already occupied".format(seat))

    self._seating[row][letter] = passenger

Notice how the call to _parse_seat() also requires explicit qualification with the self prefix.