Sets really came in handy for this challenge, as did recognizing that you can use powers of two to compute the points for each card. I tried using a regular expression to parse each card, but ended up just doing it manually with split :|

Numbers = set[int]
Card    = list[Numbers]

def read_cards(stream=sys.stdin) -> Iterator[Card]:
    for line in stream:
        yield [set(map(int, n.split())) for n in line.split(':')[-1].split('|')]

def main(stream=sys.stdin) -> None:
    cards  = [numbers & winning for winning, numbers in read_cards(stream)]
    points = sum(2**(len(card)-1) for card in cards if card)
    print(points)

This took me longer than I wished... I had to think about it carefully before seeing how you can just keep track of the counts of each card, and then when you get to that card, you add to its copies your current count.

def main(stream=sys.stdin) -> None:
    cards  = [numbers & winning for winning, numbers in read_cards(stream)]
    counts = defaultdict(int)

    for index, card in enumerate(cards, 1):
        counts[index] += 1
        for copy in range(index + 1, index + len(card) + 1):
            counts[copy] += counts[index]

    print(sum(counts.values()))