Add recorder engine name and version to analytics

[ludeeus]

Proposed change

For those who have chosen to share their usage data, we're now extending our analytics integration to include the name and version of your configured database.

Requires home-assistant/analytics.home-assistant.io#688

Type of change

• Dependency upgrade
• Bugfix (non-breaking change which fixes an issue)
• New integration (thank you!)
• New feature (which adds functionality to an existing integration)
• Deprecation (breaking change to happen in the future)
• Breaking change (fix/feature causing existing functionality to break)
• Code quality improvements to existing code or addition of tests

Additional information

• This PR fixes or closes issue: fixes #
• This PR is related to issue:
• Link to documentation pull request: Add recorder engine to analytics documentation home-assistant.io#26183

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[home-assistant]

Hey there @home-assistant/core, mind taking a look at this pull request as it has been labeled with an integration (analytics) you are listed as a code owner for? Thanks!

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[epenet]

I wonder: should we also have the engine version? see #87741

[emontnemery]

Looks good, but I agree we should consider adding the engine version too