Regarding the companion app (not bungie.net)... I had the same problem on my tablet (Android) until a friend pointed out that the character controls were moved to the bottom (they were top-left for Destiny 1). Look along the bottom of the app for (I think) 5 icons. The middle is for gear. Once selected, you can now select your character (or vault) from a pull-down at the top of the app. As for the web site app, still looking for it (I don't think it exists).

Fixed. Download destiny 1 trial on new console, agree to a character transfer before start playing and voila