Shameless plug: I developed a similar project that is Web-based (source [1], demo [2]) and supports having multiple connections configured. Both github.com and GitHub Enterprise are supported. Other backends (e.g., GitLab, Azure DevOps) could be added in theory. I started this because at work, we use no less than 2 instances of GitHub Enterprise, in addition to github.com.
It is a client-side application, meaning that it is very simple to host (no backend required).
Slightly related question - is there any self-hosted github API compatible solution? I’m pretty sure the answer is no, but figured I ask.