I don't see any issue with this at all. I noticed it the other day and thought it was convenient. The same way I let the agent tag the PRs/commits with its name. I want to know what was used, in fact I wish it would identify the model used.
I'm not embarrassed by my LLM usage and I want to provide a full audit trail whenever possible to aid myself in the future.
PS: Claude archives old conversations on your computer after 30days. I've started archiving mine to the side as well for potential future use.
I'd like to know what was used but "Co-authored by Claude Code" doesn't do anything except marketing for Anthropic.
The sessions are definitely more useful but only if I'm using a single agent from a single provider.
It has strong "Sent from my iPhone" vibes.
I use many tools to develop software and if my IDE would silently start adding "This commit was created with IntelliJ IDEA Ultimate" I'd find it intrusive and obnoxious, too.