The blue dot showing your location is only available in the iOS and Android apps. The website and the apps work on different platforms. While the app can determine the precise location because the device has this capacity, we do not show the precise location on the website.
To show your location on the app, enable Location Services (and Precise Location) for the Flightradar24 app from your device settings and go to Flightradar24 app > Settings > Map > Show my location and make sure this option is enabled.
On the web browser, you can center the map by your browser's location by pressing the arrow icon on the right side of the map. See the screenshot below.
You can also save your favorite locations using the Bookmarks widget. You can read more about Bookmarks here: https://www.flightradar24.com/blog/say-hello-to-the-new-bookmarks-widget/