Are they?
I think they are not.
Gradient descent is guided toward the minimum cost solution by following the mathematical gradients of the training data.
A random walk is simply random.
To an observer who didn’t know the system or task the weight updates could look like a random walk.
There is also random hill climbing where you make some random changes to the weights, if that improves the performance of the network, then keep those changes, else reject them. That is just a bad way to train a neural network, surprisingly not a terrible way. There are cases where you don’t really have proper information to back-propagate (eg. delayed reward situations.) Then hill climbing or evolution can be used.