Register agent identities behind use_agent_identity (#17386)

## Summary

Stack PR 2 of 4 for feature-gated agent identity support.

This PR adds agent identity registration behind
`features.use_agent_identity`. It keeps the app-server protocol
unchanged and starts registration after ChatGPT auth exists rather than
requiring a client restart.

## Stack

- PR1: https://github.com/openai/codex/pull/17385 - add
`features.use_agent_identity`
- PR2: https://github.com/openai/codex/pull/17386 - this PR
- PR3: https://github.com/openai/codex/pull/17387 - register agent tasks
when enabled
- PR4: https://github.com/openai/codex/pull/17388 - use `AgentAssertion`
downstream when enabled

## Validation

Covered as part of the local stack validation pass:

- `just fmt`
- `cargo test -p codex-core --lib agent_identity`
- `cargo test -p codex-core --lib agent_assertion`
- `cargo test -p codex-core --lib websocket_agent_task`
- `cargo test -p codex-api api_bridge`
- `cargo build -p codex-cli --bin codex`

## Notes

The full local app-server E2E path is still being debugged after PR
creation. The current branch stack is directionally ready for review
while that follow-up continues.

codex-rs/tui/src/local_chatgpt_auth.rs

@@ -108,6 +108,7 @@ mod tests {
                 account_id: Some("workspace-1".to_string()),
             }),
             last_refresh: Some(Utc::now()),
+            agent_identity: None,
         };
         save_auth(codex_home, &auth, AuthCredentialsStoreMode::File)
             .expect("chatgpt auth should save");
@@ -154,6 +155,7 @@ mod tests {
                 openai_api_key: Some("sk-test".to_string()),
                 tokens: None,
                 last_refresh: None,
+                agent_identity: None,
             },
             AuthCredentialsStoreMode::File,
         )