[FEATURE] VS Code extension to support VS Code workspaces

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

Working in a VS Code workspace via the Claude code extension is kinda broken.

When your workspace consists of multiple folders, the project that folder that gets stored inside ~/.claude/projects is the first entry from the code-workspace file. Example:

We have a workspace file stored in ~/Developer/ProjectXWorkspace with content:

projectX.code-workspace:

``` json
{
"folders": [
{
"path": "fodler1",
},
{
"path": "folder2",
},
],
"settings": {},
}

In `ProjectXWorkspace` we have 2 project folders: `fodler1` & `fodler2`

When conversation happens via the VS Code Claude Code extension this folder appears under `~/.claude/projects` -> `-Users-<MACuser>-Developer-ProjectXWorkspace-fodler1`. The project name stored depends on the path of the first folder in the workspace file.

When one uses the CLI Claude Code, this behaviour is different. The project that gets stored under `~/.claude/projects` is `-Users-<MACuser>-Developer-ProjectXWorkspace`

If VS Code Extension was working as the CLI version, we can easily switch between CLI and Extension when working on a given workspace.

### Proposed Solution

Make Claude Code VS Code extension produce the same artifacts as the CLI Claude Code when working with VS Code workspaces.

### Alternative Solutions

_No response_

### Priority

Critical - Blocking my work

### Feature Category

Other

### Use Case Example

1. I use VS Code to open multiple project dirs and save them as a VS Code workspace;
2. Claude Code VS Code extension is spawned and a conversation is made;
3. I want to open the same conversation in Claude CLI, but can't as the session that gets generated is not in the right project and I can't resume it.

### Additional Context

_No response_