This calls for the same sollution as setting difficulty level, choosing stage and so on. Generally You need to design your scene tree structure in a way, that there is some menaging or data-keeping structure above the actual platformer and everything else. In project runtime You will find yourself changing scenes, returning to menus, showing victory/gameover screens and so on. Because of this it is impossible to keep all the data in platformer itself, higher scope variables will be needed. You can use Autoload for this, or customly made node, that will be root of your scene.
How it works will be : choose character in menu -> go to character selection menu-> clicked character data is passed from selection menu to Autoload/root-> start the game ->when level loads it reads character data from Autoload/root -> done