Session identity doesn't surface which physical machine it's bound to

A Claude Code/Desktop session is bound to the machine it was started on (its shell, filesystem, and process all live there) — but the session list is synced and shown identically across every machine the user is logged into. Nothing in the UI surfaces which physical machine a given session is actually running on, so a user working across multiple machines has to remember or guess where a session lives before typing into it, especially confusing when two sessions have similar names/purposes.

Related: when Claude gives the user a command/instruction to go run somewhere, it's often unclear which machine that's meant to be executed on — the one the current session is bound to, or another one being discussed/referenced in the conversation. This ambiguity compounds the session-identity problem above.

Suggestion: show the originating hostname/machine (and maybe online/offline reachability) directly in the session list/title bar — something like the environment block already reported internally (platform, hostname) surfaced in the UI, not just to the model. When Claude hands the user a command to run, it should also be clearer/explicit about which machine that command targets.