Are there any chips which manages supercap charging and releasing power?
Yes there are ICs out there which do what you want.

They are usually labelled as backup controller or something like that. Some charging controllers will do what you want as well.

What could be missing is the monitoring aspect. I haven't seen many which give feedback about the state of charge of the backup. There are quite a few monitoring ICs out there, so you might need a combination of two.