[FEATURE] Shared memory / persistent identity across Claude sessions

### Preflight Checklist

- [x] I have searched [existing requests](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20label%3Aenhancement) and this feature hasn't been requested yet
- [x] This is a single feature request (not multiple features)

### Problem Statement

I run multiple Claude instances simultaneously — Claude Code on my dev laptop, Claude Code on a Rocky Linux workstation (via TeamViewer), and occasionally claude.ai for document work. All of them are working on the same project.

Each instance is completely isolated. When Claude on my Rocky box spends 2 hours helping me get a Solaris 8 SPARC emulator running, Claude Code on my laptop has no idea it happened. I have to manually relay context: copy-paste outputs, share files, explain what the other session accomplished. I'm the integration layer between multiple instances of the same AI.

### Proposed Solution

1. **Shared memory layer** — scoped to the user's account. When one Claude session learns something, other sessions can access it. Opt-in, user-controlled.

2. **Account-level identity** — Claude Team already has account verification. Use that as the privacy/access boundary. No cross-user leakage, just continuity within my own account.

3. **Persistent identity** — with shared memory and account scoping, Claude effectively becomes one continuous assistant rather than N independent stateless instances.

### Alternative Solutions

In my opinion there aren't any alternative solution. It must be done.

### Priority

High - Significant impact on productivity

### Feature Category

CLI commands and flags

### Use Case Example

I maintain legacy semiconductor process control systems (Solaris/SPARC, Windows XP, Fedora, Rocky Linux). In a single week I might:

- Use Claude Code on my laptop for SOW documents, memory tracking, and codebase work
- Use Claude on a Rocky Linux box to set up QEMU emulation and TeamViewer automation scripts
- Use Claude on claude.ai for quick document drafts
-

### Additional Context

## Why This Matters

The `.claude/memory/` system in Claude Code is a creative workaround, but it only works within one machine's Claude Code sessions. It doesn't bridge to claude.ai or to Claude Code on a different machine. The manual context-bridging overhead is real and scales with the number of concurrent sessions.

Users who work across multiple machines and multiple Claude interfaces — which is increasingly common — would benefit enormously from this.

The issue names .claude/memory/ as the current single-machine entry point; start by reviewing how that memory is scoped and how Claude Code sessions are identified. Then trace the CLI commands and flags mentioned in the request. Done would mean an opt-in, account-scoped memory and identity path that works across concurrent Claude Code sessions and claude.ai without cross-user access.

Do mô hình lập chỉ mục viết ra từ nội dung của issue.