Obsidian can have differing use cases on a per-device basis.
E.g. let’s say I journal on my desktop, but I keep my grocery list for use on my mobile phone.
Proposed solution
Give Bookmarks its own sync setting.
Current workaround (optional)
Maintain a note that has a list of links with headings for each device that are the ‘bookmarks’ for that device
A workaround for this is to disable Settings > Sync > “Core plugin settings”. This will affect other things, but I believe for most people they are things they don’t change the settings of once they’re setup.