Unfortunately, Site Kit by Google isn't for Shifter, because the Site Kit detects running docker's URL (e.g. abc-123-defg.app.getshifter.io) while setting it up on WordPress and you cannot change URL to your own domain (e.g. www.example.com) after setting up completed. So we want to introduce some alternatives.
A. Search Console
Add your web site's URL to Google Search Console.
It would be best if you vilified ownership with the blow:
HTML tag
Google Analytics
Google Tag manager
Domain verification
Note:
Search Console will not appear on the WordPress dashboard on Shifter, you might check it on Search Console web page.
B. Google Analytics and Google Tag Manager
The below WordPress plugins will help you to insert code provided from Google Tag Manager to the head element.
Also, Google Analytics will not appear on the WordPress dashboard on Shifter; you might check it on Google Analytics web page.
C. Google Adsense
Use the same plugins in the above to insert ads anywhere you like.
Adsense report will not appear on the WordPress dashboard on Shifter; you might check it on Adsense report web page.
D. PageSpeed insights
The page speed insights will not work for running WordPress on Shifter directly. You can test it on running WordPress with GoogleChrome's extension: Lighthouse; or run PageSpeed Insight to the generated artifact.