[FEATURE] Plugins installed via claude.ai propegate to the Claude Code CLI

### 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

https://github.com/anthropics/claude-code/issues/45323#issuecomment-5528872620 was closed. It says to reopen if this is still an issue, and it is still an issue.

When I install a connector in claude.ai, I get that connector in claude.ai, claude desktop (chat/cowork/code) and the claude code CLI.

When I install a plugin in claude.ai, I get that plugin in claude.ai, claude desktop (chat/cowork/code) BUT NOT the claude code CLI. This is annoying. It would be nice to enable a plugin across all claude surfaces at once. I keep having to remember to manually install my plugins in claude code, across all my machines.

If my claude.ai connectors can propagate to my claude clode cli, my claude.ai plugins should propagate too!

### Proposed Solution

Plugin marketplaces and plugins installed via claude.ai propagate to claude code cli.

### Alternative Solutions

Remember to manually install the plugin in claude code on every machine. This is annoying.

### Priority

Medium - Would be very helpful

### Feature Category

CLI commands and flags

### Use Case Example

I have a personal repository of skills. I bundled that repository into a plugin, so I can easily install all of my skills at once. I setup my repo as a plugin marketplace, and installed the marketplace and plugin on claude.ai.

This is great! I now have access to all my skills, across claude.ai and claude desktop!

However! I do not have all my skills in the claude code cli! I need to manually install them there, on every machine I use. Sometimes I forget, and then I get frustrated when I'm in the middle of my work and realize my skills are not there.

### Additional Context

TBH itd be really nice to be able to globally define my config and customization in claude.ai and have it all propagate to all claude surfaces (claude.ai, claude desktop, claude code cli).

It is confusing that some settings/customizations only affect the web, some settings/customizations propagate to claude desktop, and some settings/customizations propegate to claude code cli too (so far only connectors I think)

All of this is linked to my account, so I would like it to propagate to everywhere I use my account!

(Also, if I have specific customizations I only want to use in a given project, I can define them locally in the project folder or git repo! Having a way to do global settings/customizations doesn't preclude me from doing local customization)

Start by reading the existing Claude Code CLI plugin installation flow and the connector propagation behavior referenced from issue #45323. The request is done when plugins and plugin marketplaces installed in claude.ai are available in Claude Code CLI for the same account, without requiring manual install on each machine.

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