It looks like there are still some incompatibilities with the odin inspector. Therefore I disabled it for most of the scriptable objects that need to use localized properties. Localized Strings seemed to work at first but I recognized a bug that is really annoying: If I select a scriptable object in the editor and add further objects to my selection afterwards, every Localized String of every selected objects gets overwritten with the values of my initial selection. In the end, every object of my selection has the same values and I have to readjust them or discard my changes via version control. This bug doesn’t occur if odin inspector is disabled for the corresponding objects.
Could you file a bug report, please?