Here is my C# solution.

In short, we calculate sizes of containers and count of balls of each type. And then, for each type of balls we try to find a container of appropriate size. If we can do this we succeed, otherwise we fail.

static bool OrganizingContainersCore(List<List<int>> containers) {
    // containers: size of container -> count of that size
    Dictionary<int, int> containerInfo = new();

    // balls
    int[] balls = new int[containers.Count];

    foreach(var container in containers) {
        int containerSz = 0;

        for(int j = 0; j < container.Count; j++) {
            int num = container[j];
            balls[j] += num;
            containerSz += num;
        }
        containerInfo.TryGetValue(containerSz, out int count);
        containerInfo[containerSz] = count + 1;
    }

    // compare
    foreach(int ballCount in balls) {
        if(!containerInfo.TryGetValue(ballCount, out int b))
            return false;

        if(b > 1)
            containerInfo[ballCount] = b - 1;
        else
            containerInfo.Remove(ballCount);
    }
    return containerInfo.Count == 0;
}