Tree structure that is like a quadtree/octree but splits a different number of times in each dimension?
You could use a k-d tree.  A k-d tree normally alternates between partitioning horizontally and partitioning vertically, but you could certainly modify it to change that pattern (e.g., partition horizontally twice in a row; or equivalently, partition horizontally into three or four regions rather than just into two regions).

Not everything has a "name".  There are many more possible algorithms and data structures than there are "names".  So don't take it as some kind of negative if you can't find a "name" for it.  "Modified quadtree" or "modified k-d tree" will do fine, if you need a way to refer to it.