You can export your bookmarks and then open them on another device. Starting from release 8.2.4, you have possibility to back up bookmarks in a cloud storage. However, automatic synchronisation is not an easy task, and it takes time, we hope, that it will be available in a longer term.
Note: the Profile tab in the app settings is used to send edits made in the map editor to OpenStreetMap.org (OSM). Your bookmarks are not backed up in the OSM profile.