Hello @fiamin,
Thank you for contacting Rank Math support.
Google uses different sources to figure out and show your Site Name on SERPs og:site_name Website structured data (Schema) Content in title elements Heading elements, like H1 Among them, Rank Math automatically includes og:site_name, meta tag and the Website Schema on your page. You can also check the Google documentation on site-name: https://developers.google.com/search/docs/appearance/site-names
However, as mentioned in the documentation, this process is automated by the algorithm, and even though it takes into account the information on the website, it is still not guaranteed that your site name will come up correctly in Google SERPs since it’s solely decided by Google what they found relevant and user-friendly and which one they want to show.
Please make sure that you have added the correct site name in WP Dashboard > Rank Math SEO > Local SEO > WebSite Name and our plugin will add it to the meta tags and Schema. Also, make sure that the Site Title is set correctly in the WordPress settings.
Once done, wait for Google to decide to show it in the SERPs.
Hope that helps and please do not hesitate to let us know if you need our assistance with anything else.