Yes, it's reflexive in this context. When someone feels something themselves, like an emotion or sensation (always an adjective), you add the reflexive pronoun as an object.
"Å føle" can also be used when you feel something else, and then uses that "something" (always a noun) as its object. It's still transitive, but not reflexive.
I translated this as "she feels like herself at home," as in she feels very comfortable and natural at home. In my translation, the focus is on the self, where it seems as if "she feels at home" places the focus on the building. Is this a correct analysis. If so, how does one say, "she feels like herself at home."?
"She feels at home." It has no focus on the building. It is an expression that means she feels comfortable and natural as if she were home. You do know that home is where the heart is. It is not a building. Your home could be in a house or an apartment or a houseboat or a tent or anywhere. In English "she feels like herself" means that she doesn't feel like someone else or that she recognizes that she acts as she usually does. We might say "She feels like herself again." after she is no longer sick or drunk or whatever made her not be like herself.
They're synonymous in this context, but "kjenne" has some additional meanings. You can "kjenne" (know) a person, for instance, and "å kjenne igjen" means "to recognise".
There's a tendency for "å kjenne" to be used more often with physical sensations, and "å føle" more often with emotions, but there's no real divide there and often the choice comes down to dialect.