AFAIK when going from one language to another you often have known transliterations. Sometimes that can be a mapping from "Ö" to "Ö".

For cyrillic a fixed transliteration is both more needed. "o" and "ö" sound different, but even if you pronounced any "ö" as "o" everybody would understand what you meant. This is different for cyrillic where there are many letters where typical western readers wouldn't even know how to pronounce them. A "Щ" would look like a weird "W", but is romanized as "shch" or "sch".

Even more extreme examples would be the romanization of japanese.