I think this is a common enough scenario in the US.
Imagine an intersection of a North/South road and an East/West road.
On the East side of the N/S road the E/W road is named something like 13th St. On the West side of the N/S road the E/W road is named something like University Ave.
Perhaps more common is the simple change of something like East Main St. to West Main St., but I'm not sure that's considered the same, or if there's a different term for that.
Does a term or phrase exist to describe this type of intersection that seems to define the cut-off points for different legal names for a continuous stretch of road?
Here's an excerpt from Wikipedia on street names, which talks about this:
Where a street crosses or forms (straddles) a boundary, its two sides sometimes have different names. Examples include Seton Avenue (Bronx) / Mundy Lane (Mount Vernon, New York); Station Road (Portslade) / Boundary Road (Hove, East Sussex); Lackman Road (Lenexa) / Black Bob Road (Olathe, Kansas); Nieuwstraat (Kerkrade, Netherlands) / Neustraße (Herzogenrath, Germany), both names meaning 'New Street'.
But it also lacks a term or concise phrase for this convention.