None of this readily translates to France. There is no mandatory registration system like in Germany but different notions of residence for different purposes so that “résidence permanente” is somewhat vague or ambiguous and I would not use it to refer to anything. I am not sure I have heard it either, at least not when referring to France (as opposed to the US or Canada, where permanent residence is something very specific).
What you will find in most legal contexts is simply the word “résident”, which will have a specific meaning depending on how it is used. If you need more details, the appropriate phrase will also depend on the context, for example “résidence principale” (Hauptwohnsitz, as opposed to a vacation home) or “résidence habituelle” (used for civil law or customs purposes).
As far as immigration law is concerned, I don't think France uses the phrase “permanent residence” either. The closest you will find to a Niederlassungserlaubnis is simply called a “carte de résident”. You don't need to add “permanent” because anything less permanent than that will have another name like “carte de séjour”, “visa de long séjour valant titre de séjour”, “récépissé de demande d'asile”, etc. Technically all these, including the carte de résident, are types of titre de séjour.
And outside of legal contexts, I am not sure many people would use the word “résidence”, it sounds very formal. If you are on holidays and want to talk about your usual place of residence, you might say “Mon adresse est […]” or “J'habite à […]”.
Incidentally, Canada does have a permanent residence programme and uses the phrase “statut de résident permanent”.