The ribbon icon order and displayed/hidden state is saved in the .obsidian/workspace.json file on desktop and in .obsidian/workspace-mobile.json on mobile.
Obsidian desktop uses only the first and Obsidian mobile uses only the second.
There may be a FR to combine them, but for now they are separate for desktop and mobile.
I have now also compared Mac and iPad, which should align according to the workspace.json. It does display the same content, but not in the same order. The whole feature somehow doesn’t seem to work properly…