It's not too squishy. The important thing to understand here is that a plane is an infinite object in a given subspace - just like a line is. Two infinite objects intersecting give you an infinite object in the subspaces they share, and a zero-sized object in the subspaces they don't share. The shared subspace of two planes in 2D is obviously the same plane. The shared subspace of two planes in 3D is obviously either zero (two parallel nonincident planes), a line (the intersection that they share) or a plane (two incident planes). So you get a null, a one-dimensional subspace, or a two-dimensional subspace, depending on how many shared subspaces you have between the objects. So a pair of three-dimensional infinite objects, in four-dimensional space, may either be disjoint, resulting in a null, incident, resulting in an infinite volume, or intersecting, resulting in an infinite plane.
The source of confusion here is that we're mixing up (bounded) volumetric shapes, line segments, and planar polygons, with their infinite counterparts - the lines, points, planes, and infinite volumes. An infinite object has no shape - it is infinite in some dimensions and zero in all others - this is what gives us those clear well-defined intersection objects, which are either zero or infinite. A bounded object has a shape - it has a boundary that is not infinite. When you intersect such objects within their shared subspaces (two line segments on the same line, two polygons on the same plane, two volumes in the same space) you get either a null, or an object of the same type. This is obvious when you make one of the objects infinite - an intersection of a line segment with its line gives you the same line segment. So the reason you are having difficulty imagining two somethings that will intersect in a plane is that a plane is an infinite object, so only two infinite objects can intersect in a plane. So you need two infinite volumes, that pass through the same 4D space, but are not the same volume. These volumes share a two-dimensional subspace - the area of 4D space they intersect in is a plane. It's infinite in two dimensions, and zero-sized in two others.