It works for me. At the drop down list from my username in the top right of a Garmin Connect page I select 'settings' and then at the 'display preference' I choose 'metric'. It saves it automatically now and there should be a little yellow highlighted note above that pops up saying preference saved.
And you are talking about the website through a browser on your PC/mac and not some smartphone app right?
I was able to change the setting to Metric via the settings-option after clicking on my user name (somewhere top right on the page). It saved and since that moment all distance, pace and speed are shown in KM.
For some reason, when I share an activity via social media everything is shown in miles again. Same goes for live tracking.