Is it possible to implement an activation function or layer in Keras that uses two distinct sets of weights?

The layer (or activation function) you described could be easily implemented. It takes two input tensors, x and y, which have compatible shapes and return as output 1 / (1 + y * e^(-x)). So using the documentation as our guide, we can implement it like this:

import tensorflow as tf

class MultSigmoid(tf.keras.layers.Layer):
    def __init__(self):
        super(MultSigmoid, self).__init__()

    def call(self, inputs):
        x, y = inputs
        return 1.0 / (1.0 + y * tf.math.exp(-x))

And now, just for demonstration, we can use it like this:

inp1 = tf.keras.layers.Input(shape=(3,))
inp2 = tf.keras.layers.Input(shape=(5,))

d1 = tf.keras.layers.Dense(6)(inp1)
d2 = tf.keras.layers.Dense(6)(inp2)

out = MultSigmoid()([d1, d2])

Note that in our implementation we assumed that y * e^(-x) is the element-wise multiplication. If instead, we are interested in tensor (or matrix) multiplication, we can use tf.matmul like: tf.matmul(y, tf.math.exp(-x)); however, note that in both cases it's assumed that the shapes are compatible with each other and the multiplication could be done, otherwise an error is raised.