Or rather, does "speed" relate to the same measure? The speed of light is the speed of a photon/an electromagnetic wave in the empty space, but gravitational waves are wave of this very same space? How to make sense of it in a more formal way?
When we talk about black holes, we usually talk about the insides of the event horizon. The escape velocity of a event horizon is more than the speed of light therefore only particle like tachyons which travel faster than light can escape, but those are hypothetical so technically nothing can escape the event horizon, and the speed of gravity, bending of spacetime will be equal to gravitational waves, ripples in spacetime. So these waves can't escape the event horizon.
But the point is that, gravitational waves don't occur from the insides, they occur outside of the event horizon. If an assymettric distribution of mass is there then the gravity of the masses, will create gravitational waves along it's barycenter, which happens outside of the black hole. Let's take an example of 2 blackhole's merging.
Both the black holes move violently around, till the merge, till the event horizon collides. After that we are unable to observe gravitational waves, we observe gravitational waves from spindown because spindown is also external to the black hole
They are just like other fields like photons, deflected by a massive object so it can also not escape.
In order for gravitational waves to occur, frame-dragging (spacetime dragged by gravitational fields, stretching it) may occur, which happens outside the event horizon (within or on or outside the ergosphere but never inside the event horizon) as the insides are not a part of our universe.
This has been tried in a thought experiment known as stick bead argument that whether gravitational waves have physical effects like this.
Note: The only restriction is Young's modulus, as expansion of universe is speeding up the spacetime is stretching more and more and making it difficult to form waves
Note: Gravity can escape a black hole and a gravitational wave cannot because the curvature of spacetime that causes gravity is not at all influenced by the mass of the object, it is the local region where the gravitational field extends so it does not care about the event horizon, when an massive object is there on spacetime, the cause of curvature isn't mass, it's the surrounding area which pulls on to it. Even if gravity were particles according to QFT, there is no way they could escape because the force carrier, graviton would be as a virtual particle and they are not at all restricted by speed. While gravitational waves are unable to escape the horizon because they are influenced by a mass (since they are bent), they are bent, deflected or even swallowed by the singularity