This is a bit strange, as one would expect to have the hand-off meeting, knowledge transition and all on-boarding process completed before officially being assigned a role. Usually, once you're "officially" assigned a role, you are expected to assume the responsibilities immediately. There are usually two scenarios:
You are actually replacing the previous person: In this case, it's absolutely necessary that you understand the role, responsibilities and is well aware of the process to be able to be functional from day 1.
You are being added into the role, while the existing person handling that role is going to transition away later: In this case, you have some buffer time to learn the role and responsibilities, however, the sooner you understand and can be functional independently, the better. You'll have less dependency.
Also, as you mentioned
I had an initial discussion with my predecessor about the role, mostly about the responsibilities involved, before I accepted it. This happened couple of months ago.
People might be under the impression that you learned all the required tips and tricks and the handover (official or not) might have taken place.
In either ways, please reach out to your superior and your predecessor as soon as possible, and ask for the hand-off meeting. There's nothing to be gained by waiting.