Google Tag Manager is a tag management system from Google. Through a web interface you configure and deploy JavaScript and HTML tags, including web beacons for web tracking and analytics, on a website or in a mobile app without editing the site's code. You can test and adjust changes in the same environment. With server-side tagging, tag processing moves to a tagging server of your own, which you set up on Cloud Run, App Engine or manually on a platform of your choice and run under your own domain. It is not open source.
Google is a US company and is subject to US law. The web container is delivered by Google. With server-side tagging you decide where the tagging server runs and which data it receives and passes on. For a European organisation, what matters most is which tags are loaded through Tag Manager and where those tags send data.