We save your swing in your iCloud account. Please make sure you have logged into your iCloud account, otherwise, saving would silently fail.
In the phone's Settings app, Your name (the first button)->iCloud->iCloud Drive must be toggled on. Then, scroll down, SwingAnalyzer must be toggled on.
On the watch, when the swing stats is shown, make sure to scroll down and click the “save swing” button. A confirmation pop up should show in a couple of seconds if the data is saved.