This feature has been asked to be added since almost 20 years ago,
here: https://bugzilla.mozilla.org/show_bug.cgi?id=444284
Currently, Firefox Sync does transfer essential data (bookmarks, history, passwords, add-ons), but it does not include search engine shortcuts or custom-added engines or even deleted search engines (google,etc).... This missing functionality creates significant friction, requiring repetitive and time-consuming manual setup, which diminishes the utility of the Sync feature. (Especially if you have search engines added with custom preferences or very specific search engines added through mycroft for example..)