Maturi0n (2019-01-24 19:42:30+00:00): Abkhaz and Russian are both official languages in separatist Abkhazia, but Russian is more widely used and it has an established method for its transcription into German or English, while such thing does not exist for Abkhaz. Transliteration is actively discouraged from being used in OSM as they can much better be generated automatically by scripts or tools.

There is no centralized authority that defines by which name a place in Abkhazia has to be referred to. I am a native speaker of German, and besides Russian, I have a high fluency in English and French and speakers of these languages definitely don't "automatically use Soviet/Georgian" names, because 90-95% of speakers of those languages probably don't know a single city or village in Abkhazia, and if they do, it's mostly Sukhumi. Just because the German or French government recognize Abkhazia as part of Georgia doesn't mean they oblige their citizens to use a Georgian-based transcription of place names in Abkhazia.

By the way, in case of German, there is considerable "proof" that, in the rare cases when Abkhazia appears in books, media, etc, often a Russian-based spelling is used. Russian is still one of the most popular foreign languages here and some citizens of the former GDR have visited Abkhazia. One place that at least a few Germans know is Pitsunda. There are several books about it ("Hochzeitstag in Pizunda", "Pizunda, die hellblaue Meeresbucht") and they all refer to it by its Abkhaz/Russian name, and don't call it "Bitschwinta", which would be the German transcription of the Georgian name.

In case of "Otschamtschyra", that name is also used by German media, for example here: https://www.mainpost.de/regional/bad-kissingen/Mein-Herz-ist-in-Otschamtschyra;art766,5948095 
However, in this case "Otschamtschira" is used more commonly, especially in academic publications.

Even "Gjatschrypsch" is specifically referred to by it's Russian-based spelling: http://www.spiegel.de/spiegel/print/d-66567972.html

You cannot compare the ties Turkey has to Akhaltsikhe, which once belonged to the Ottoman Empire and was mostly inhabited by Turks (Meskhetians) until the mid 20th century to the ones Germany and France have had to Abkhazia. They were and still are almost zero. The average German or French probably doesn't even know Abkhazia exists. No speaker of those languages automatically switches to the Soviet or Georgian name when referring to a place in Abkhazia because he doesn't know it. Nor does he know the current Russian or Abkhaz name. 

The point here is, that while both the Georgian/Soviet and the Abkhaz/Russian names are completely unknown to almost every speaker of those languages, the old Georgian/Soviet name actually has zero practical use in Abkhazia for a speaker of those languages. They are not used on signs anymore, locals younger than 35 probably have never heard of the old names, older people might even react with anger in some cases.
If a speaker of German sees "Gjatschripsch" on the map, it will allow him pronounce the name currently used on the ground, even if he doesn't read Cyrillic. "Lesselidse" will get him nowhere and would be in contrast to the name used on the ground.
For example Stepanakert, capital of Nagorno-Karabakh, is also refered to by it's Armenian-based spelling and not as "Xankəndi" in German despite the German government also not recognizing the separatist government there.