Similarly in an ambient 3-d space, 2 generic planes intersect to give a line. But the planes can also be coincident or parallel. And in higher ambient spaces can intersect at only a point, rather than a line, or even fail to intersect in a non-parallel way.
In 4-d space, the intersection of two generic 3-d spaces does indeed give a plane, with exactly similar caveats.
The standard GA doesn't directly represent general lines or planes, however. The elements are the equivalent of "vectors" rather than "points", and always go through 0. The obvious way to handle these are parameterizing the lines and surfaces, but you're essentially working with equations for the surfaces, and keeping track of the variables.
The slick way of handling it is with _projective_ geometric algebra, and intersections turn into "meets". The meet of two parallel lines (planes) is now a "point (line) at infinity", and of a (line, plane) with itself is the line (plane) again. Skew lines have a meet of 0 (not the point 0, the number 0).
GA has been well discussed here, so, the other half:
Projectivization is a fairly standard trick even for normal geometry. It's adding an additional dimension, which is in most contexts just set to 1. (Projective actually just means treating all points on a ray as equivalent; this loses the dimension you just gained
It lets rotations and translations be treated in a nearly uniform manner, and lets you do rotations around points that aren't the origin. It's used all over the place in much graphics code (usually under the name homogeneous coördinates). The last section of https://en.wikipedia.org/wiki/Homogeneous_coordinates discusses this briefly.
For online resources for the combination: another commenter has recommended https://bivector.net/ , and it looks okay, with sections specifically on 2 and 3 dimensional projective geometric algebra.
There is also the nice C++ header library klein: https://www.jeremyong.com/klein/
Two 3D volumes in a 3D space obviously intersect to give another 3D volume, unless they're tangent. So at first glance I'd be tempted to say no, but...
Two overlapping polygons in a 2D space also define another 2D polygon. Two overlapping line segments in a 1D space define another line segment. You only get a reduced-dimension object at the intersection if they're intersecting in a higher dimensional space. Otherwise they can't intersect "at an angle".
So I'd say purely by following the pattern that it must be that two 3D spaces intersecting "at an angle" within a 4D space define a plane by their intersection.
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.
Thanks for the more thorough explanation!