It's only spherical in a Schwarzschild (non-rotating) black hole. A rotating black hole is called a Kerr black hole, and stuff gets weird, such as there being an oblate event horizon, a weird outer horizon called an ergosphere where spacetime gets dragged along such that it's impossible to stand still and you can accelerate objects using the black hole, a weirder inner horizon called the Cauchy horizon where time travel is possible, and a singularity in the shape of a ring. Your intuition is correct that during a merger it would be weirder still.
Edit: Updated the bit about about horizons as I research a bit more. It's complicated, and I'm still not positive I have it exactly right, but I think it's now as good as I can get it.
https://en.wikipedia.org/wiki/Kerr_metric
https://arxiv.org/pdf/0706.0622
https://en.wikipedia.org/wiki/Ergosphere
https://en.wikipedia.org/wiki/Cauchy_horizon
Edit: Updated the bit about about horizons as I research a bit more. It's complicated, and I'm still not positive I have it exactly right, but I think it's now as good as I can get it.