How to apply class weight to a multi-output model?
I wansn't able to use the class_weight parameter yet, but in the mean time i've found another way to apply class weighting to each output layer.  

Current solution

In this keras issue they have supplied an easy method to apply class weights via a custom loss that implements the required class weighing.

def weighted_categorical_crossentropy(y_true, y_pred, weights):
    nb_cl = len(weights)
    final_mask = K.zeros_like(y_pred[:, 0])
    y_pred_max = K.max(y_pred, axis=1)
    y_pred_max = K.reshape(y_pred_max, (K.shape(y_pred)[0], 1))
    y_pred_max_mat = K.cast(K.equal(y_pred, y_pred_max), K.floatx())
    for c_p, c_t in product(range(nb_cl), range(nb_cl)):
        final_mask += (weights[c_t, c_p] * y_pred_max_mat[:, c_p] * y_true[:, c_t])
    return K.categorical_crossentropy(y_pred, y_true) * final_mask


where weights is a CxC matrix (where C is the number of classes) that defines the class weights.
More precisely, weights[i, j] defines the weight for an example of class i which was falsely classified as class j.

So how do we use it?

Keras allows to assign a loss function for each output.
so we could assign each output a loss fucntion with the correct weights matrix.  

For example, to satisfy the request i made in the question we could suggest the following code.

# Define the weight matrices
w1 = np.ones((2, 2))
w1[1, 0] = 10
w1[1, 1] = 10

w2 = np.ones((3, 3))
w2[0, 0] = 5
w2[0, 1] = 5
w2[0, 2] = 5
w2[2, 0] = 10
w2[2, 1] = 10
w2[2, 2] = 10    

# Define the weighted loss functions
from functools import partial
loss1 = partial(weighted_categorical_crossentropy, weights=w1)
loss2 = partial(weighted_categorical_crossentropy, weights=w2)

# Finally, apply the loss functions to the outputs
model.compile(loss={'output1': loss1, 'output2': loss2}, optimizer='adam')


And that accomplishes the request :)

Edit

There is a small edition that needs to be made.
The loss functions must have a name, so we can supply this with the following:

loss1.__name__ = 'loss1'
loss2.__name__ = 'loss2'