You can not wizard never save any thing automatically in the database.
It is the responsibility of the transition to store if needed the data from the view.
Also for Many2Many field, the wizard will only use the list of ids so you must use a One2Many field if you want to have the content.
But in general wizard are not the right tool to update stored records, you should directly edit the stored record in a form or use Model to collect the modification to apply to a record.