"Hast du eine eigene Wohnung?"
I think "Das ist deine eigene Wohnung" is all right, but "Hast du deine..." sounds odd, just like "Do you have your flat?"
I think you missed something in your English sentence. I think that "Hast du deine eigene Wohnung" would be translated as "Do you have your own flat". In that case, both English and German seem fine by me.
I missed "own" on purpose to show a situation when it looks awkward in English too. "Hast du deine eigene Wohnung" does not seem right to me because "eine eigene" is possible in German. The same is in Russian, there is another option to say "your own", and the literal translation from English will look funny. Languages are different.
Sorry, I didn't mean that I based my opinion on what you can do in English. I commented to say that I think that "Hast du deine eigene Wohnung" is acceptable in German. I thought you were missing a word in your English sentence and that your opinion was influenced by that.