Description of problem:

Backstage search component emits the Search analytics data such as `search`, `discover` events with additional metadata but the custom search component in global header does not emit the search analytics data.

Prerequisites (if any, like setup, operators/versions):

Setup analytics in the application.

/packages/app/src/apis.ts

createApiFactory(analyticsApiRef, {
 captureEvent: (event:AnalyticsEvent) => {
   console.log('Captured event:', event);
  },
}), 

Steps to Reproduce

1. Use search bar and see if the events are logged in the console.

Actual results:

`search` and `discover` events are not emitted.

Expected results:

`search` and `discover` events should be emitted like default backstage search component.

Reproducibility (Always/Intermittent/Only Once):

Always