[Dashboard][Managed Content] Maps and saved search refer to the same saved object after duplicating managed dashboard

[mbondyra]

Kibana version:
main

1. Create a managed dashboard (you can use an object from [Dashboard] Keep managed panels out of unmanaged dashboards #176006)
2. Duplicate it
3. Behavior: Maps and saved search are referencing the same object as from managed dashboard – now user can edit the objects they should not

It's a regression coming from the fact that our managed content tests were skipped for a few months now: tested here

Here's the code that caused a regression:
Maps: https://github.com/elastic/kibana/pull/178158/files#diff-550e794ceab85d42296fe9a224b06e66d41f036863b636565ab6d76f73d85572L227
Saved search: 5c4eae1#diff-3baaeaeef5893a5a4db6379a1ed888406a8584cb9d0c7440f273040e4aa28166L121

We have to fix it asap, otherwise we'll have to also fix it for other embeddables as the refactoring happens.

This should be included in this PR: #180938, but somehow it doesn't work correctly (https://github.com/elastic/kibana/pull/180938/files#diff-f97c836c673c4de4b70b86ed593cca242d90956d9850ee96b124f6d46587ec32L285)

@elastic/kibana-presentation I've assigned it to you as it's related to embeddable 🙏🏼

[elasticmachine]

Pinging @elastic/kibana-visualizations (Team:Visualizations)

[elasticmachine]

Pinging @elastic/kibana-presentation (Team:Presentation)