[Search Sessions] Search session inspector

[Dosant]

Describe the feature:

Part of #61738

Search sessions are a mechanism that allows to store search results and restore them.
e.g. loading dashboard using previously-stored searches

Searches that run inside the app are grouped under a search session.
We should introduce a new type of inspector: search session inspector.

Describe a specific use case for the feature:

We need a search session inspector to debug search sessions on a client.
It will also make search session integration into other applications easier.

[elasticmachine]

Pinging @elastic/kibana-app-services (Team:AppServices)

[vadimkibana]

Thank you for contributing to this issue, however, we are closing this issue due to inactivity as part of a backlog grooming effort. If you believe this feature/bug should still be considered, please reopen with a comment.