Matomo (ex Piwik) does have an "importer" for search keywords (from GA and Bing and maybe something else) but it's not free: https://plugins.matomo.org/SearchEngineKeywordsPerformance
It uses the APIs of the search engines to download/import the search keywords.
Re:Migrating Away from Google Analytics
by markosaric on Sunday July 12, 2020 @05:06AM
(#23810007)
Yeah, Plausible has a Google Search Console integration so you can see all the keyword queries people find your site with directly in the Plausible dashboard.
Do other analytics services also provide this feature? If so, I'm curious of how the manage to do it without being integrated with Google.