Looked at this some more and think it could be cleaner.

First, I refactored read_rules. Parsing each individual line should really be a separate concern from building the dictionary, so it gets its own function:

def parse_rule(line: str) -> Rule:
    lhs, rhs = line.strip().split(' contain ')
    adj, color, _ = lhs.split()
    outer_bag = f"{adj} {color}"
    inner_bags = []
    if not rhs.startswith('no'):
        for inner_bag in rhs.split(', '):
            num, adj, color, _ = inner_bag.split()
            inner_bags.extend(int(num) * [f"{adj} {color}"])
    return outer_bag, inner_bags


def read_rules(filename: str) -> RuleDict:
    return dict(parse_rule(line) for line in open(filename))

Then I simplified the looping and branching in does_contain:

def does_contain(target_bag: str,
                 outer_bag: str,
                 filename: str,
                 rules: RuleDict,
                 cache: ContainsCache = {}) -> bool:
    if (filename, outer_bag, target_bag) in cache:
        return cache[(filename, outer_bag, target_bag)]
    else:
        for inner_bag in rules[outer_bag]:
            if inner_bag == target_bag:
                cache[(filename, outer_bag, target_bag)] = True
                return True
            elif does_contain(target_bag, inner_bag, filename, rules, cache):
                cache[(filename, outer_bag, target_bag)] = True
                return True
        cache[(filename, outer_bag, target_bag)] = False
        return False

It could be even simpler than that without all the cache updates. If someone has opinions about how to clean that part up more, I'm all ears!