There is a street and U-Bahn station in Vienna called
Handelskai. Over the speakers in subway and busses I always hear it pronounced like "Handelski". However, other "Kais" (eg. "Franz-Josefs-Kai") were pronounced "kai" / "kay". Why is this only word pronounced different?
As the voice of Vienna public transport has changed two years ago, I don't know which you are referring to, but you may convince yourself by listening to the old announcements of Franz Kaida (http://www.fpdwl.at/forum/downloads/ansagen/u-bahn/6-stationsansagen-linie-u6/index.html) that he didn't say "kee" (English pronunciation) but an elongated "kay" which may sound like "kee" in the ears of an English native speaker – or a Polish one, which I can't tell. If you're not referring to the "old" voice, the new one of Angela Schneider is even better trained and certainly didn't let wrongly slip an "ee".
You can get the answer why "Kai" is pronounced like the French "quai" from etymological sources like the Deutsches Wörterbuch von Jacob und Wilhelm Grimm: "Kai" is, until today, often taken for a word of French origin and thus pronounced, although it is of Netherlandish origin ("Man findet in deutschem text häufig die franz. form, weil man das wort gewöhnlich für ein franz. hält, auch franz. ausspricht, schwerlich mit recht"). But this seems to be the case not only in Vienna or Austria, but also in several regions throughout Germany. Wherever the word has the French spelling outside French-speaking countries, of course, it is usually also pronounced that way.
In Vienna, Kai is pronounced [keː] instead of [kaɪ̯].