Sync: Widget import changes from the core

[Mamaduka]

What?

Sync changes from WordPress/wordpress-develop#3893, to avoid registering sidebars twice.

• Changes registration method and theme mod name.

Testing Instructions

See #45509

1. Active a classic theme.
2. Insert widgets into the sidebars.
3. Switch to the block theme.
4. Open a template in the Site Editor.
5. Add a new template part to the template.
6. Import widgets.

[github-actions]

Flaky tests detected in ef91644.
Some tests passed with failed attempts. The failures may not be related to this commit but are still reported for visibility. See the documentation for more information.

🔍 Workflow run URL: https://github.com/WordPress/gutenberg/actions/runs/4124050729
📝 Reported issues:

• [Flaky Test] should be able to create an hierarchical post without title support #36939 in specs/editor/plugins/custom-post-types.test.js
• [Flaky Test] should navigate block hierarchy using only the keyboard #47640 in specs/editor/various/block-hierarchy-navigation.test.js