See it yourself, this is the gulf relation: https://www.openstreetmap.org/relation/9326283

It has more than 90 names, on different languages. As I can see the default name key without a language subkey is the same as name:ar which is the Arabic name, so that's the "default". There is also an alt_name:ar as "alternative name" and google translate says it means "Gulf of Basra" which should be coming from the city of Basra, in Iraq (OSM).

In Persian the same 2 keys exist. The default name:fa translates to Persian Gulf, the alt_name:fa is Arabian Gulf

It's important to note that the language independent name tag is not used the same way everywhere. E.g. here they use the Arabic name, but for example for the Danube river this default is "Danube" in English, while for example this section where it's the border between Slovakia and Hungary its name is "Dunaj / Duna", containing names in both languages. Similar here, but Romanian and Bulgarian names "Dunav/Dunărea": https://www.openstreetmap.org/way/42369355

This tagging scheme is also common for example in Alto Adige/Südtirol (OSM) a region of Italy where 70% of the population is native German speaker, they add both names, and you can see this on the default Mapnik layer on openstreetmap.org

OpenStreetMap is a database, and the map you see on openstreetmap.org is just one renderer, a lot others exist. If you want to see everything rendered in another language you can use for example on Wikimedia Maps, in this url just change the language code at the end to something else, e.g. this is in German: https://maps.wikimedia.org/?lang=de This is in Italian: https://maps.wikimedia.org/?lang=it

Check the community discussion thread (linked in the OP). I'm not sure they cover the Persian/Arabian Gulf specifically, but in general terms your question is likely answered in that thread.

TL;DR: There are various tags, prefixes, suffixes, whole nine yards for documenting all the different types of names.