It's best to think of the map at openstreetmap.org as a demo, of the OSM database and of some of the mapping technologies that have arisen along with it -- map rendering, map search, routing.
In this case, the right-click "Show Address" feature demonstrates the reverse geocoding capabilities of the search tool Nominatim.
"Reverse geocoding" means taking map coordinates and attempting to deduce the address. In its simplest form, it does *not* look at any of the data tags of the building, but instead just tries to make a best guess based on the coordinates. In this implementation it *does* try to pull some data from the building (the building name and housenumber) but takes the city name strictly from the coordinates, which, as Maxerikson mentioned, are outside the city limits of Batavia.
My guess is that it assigns the city name Five Corners since it's the nearest placename that does not have distinct boundaries -- it' it's just indicated by a node. Nominatim "knows" the building isn't in Batavia so it's looking for the nearest place it *might* be in.
I can see how this might not look like ideal behavior. Nominatim, like everything else in the OSM world, is under constant development, and you could open a issue at the [Nominatim Github page](https://github.com/openstreetmap/Nominatim) to suggest different behavior. Don't hold your breath, though, because there may well be a very good reason for this design choice.
The good news is that the city is tagged correctly on the building, so if you want to see the house with the correct address just use the link [https://www.openstreetmap.org/way/182393593](https://www.openstreetmap.org/way/182393593).
Btw, it looks like as of now there's still one incorrect address tag, `addr:province=Batavia`. Best to delete that.