Not really, the closest analogy: Makefile or .editorconfig - it's created for a tool, committed on purpose.

What .editorconfig has in common with CLAUDE.md and similar files, is that they should be shared among the team.

Individual guidelines can live in a dot-prefixed subdirectory or the users $HOME.

This is what they already provide.

So it's possible (but not necessary) to add personal pre-prompts, but the shared best-practises and, more importantly, learnings, can be comitted and pushed.

What's exposed on a web server is a different topic, and when that is your repo's root dir, you've done something wrong either way.