The non web version uses whatever credentials you have setup yourself, so it works just fine.

Is that host level or can I provide scoped tokens based on what I'm doing?

In other words, do Anthropic tools provide any affordances for this or is it something I have to manage externally?

I'm just talking about the version of Claude Code which runs in containers on their infrastructure here - they call it "Claude Code on the web" (terrible name) and you access it through their native apps or from https://claude.ai/code

That product only works against one GitHub repo at a time. The Claude Code you install and run locally doesn't have a GitHub attachment at all and can run against whatever you checkout yourself.

Here's an open feature request about it: https://github.com/anthropics/claude-code/issues/23627