Hi Anne,
The plugin is not yet compatible with the gtag version of the tracking code, only the universal analytics.js implementation as described in the documentation.
The search statistics database (on the statistics menu) is only internal, that information is not sent, and cannot be sent to google analytics unfortunately.
The main difference between gtag and analytics.js is, that gtag is more advanced and focuses on interactions mostly, not pageviews. A solution for gtag is coming soon of course, but currently only the universal tracking is available.
The statistics page is also under re-work, as it is one of the oldest features within the plugin now. I am planning to add details about each phrase, daily, weekly, monthly analysis and similar features.
I hope this helps!