Thanks for raising the birthday paradox problem.
The search panel should show all the notes with the same code, so hopefully even with collisions you'd still be able to tell from context which note you're looking for.
I did consider allocating IDs, but I'm a bit worried about this tying notes to a specific Obsidian vault - with the way note codes work now, there is no dependence on the vault itself, only on the note's name. Something to think about...
But is it not then the same thing, because you already "allocated an ID" by naming it? Did you, before coming up with the 4-char hash, often label things outside the vault with the same names as those found in your vault?
Then it is stable, not tied to the vault, not tied to the name, can be customized on a per note basis if a user wants (00-00 is a specific note for example), and can be deduplicated within a vault.
One other possibility is use git log --follow to grab the original filepath + commit hash, to base the code on. Imperfect but should reduce code changes vs. on rename.
I really like the simple 4 letter code for shorthand!
It's an interesting problem to think on and also reminds me of resource fork advantages. My two cents is it's worth having a setting for "more permanent" codes at whatever compromise it forces. The idea of very short note codes is really nice, especially if it's "that's the code, it won't change" (I constantly rename and move notes to improve organisation and I expect others do) — I'll be trying out your plugin soon!
The Git idea was to track the original commit of note, assuming the vault is in a repo, and use that original path + git hash as the thing that doesn't change. Log tracking is imperfect, but I'd expect it to work in a vast majority of cases in Obsidian context.