If I understand your use case correctly you are dealing with two “labels”, each consisting of 10 classes.
Each sample belongs to one particular class of each label.
I think in that particular use case you could use two linear layers, one for each label, and return these two outputs:
class MyModel(nn.Module):
    def __init__(self, num_classes1, num_classes2):
        super(MyModel, self).__init__()
        self.model_resnet = models.resnet18(pretrained=True)
        num_ftrs = self.model_resnet.fc.in_features
        self.model_resnet.fc = nn.Identity()
        self.fc1 = nn.Linear(num_ftrs, num_classes1)
        self.fc2 = nn.Linear(num_ftrs, num_classes2)

    def forward(self, x):
        x = self.model_resnet(x)
        out1 = self.fc1(x)
        out2 = self.fc2(x)
        return out1, out2