Normally, yes: the plane of focus is parallel to the sensor/film plane, unless you are using a tilt lens. So if only a horizontal or vertical band appears sharp, the usual cause in macro is that the subject plane is not perfectly parallel to the camera, and at macro distances the depth of field is extremely thin.

That means tiny distance differences across the subject can make one strip look sharp while nearby areas fall out of focus. Stopping down to a higher f-number increases depth of field and can help, though macro still remains very unforgiving.

Less commonly, lens aberrations can contribute. Field curvature can make the sharpest area curve instead of lying on a flat plane, and astigmatism can make detail in one direction focus differently from perpendicular detail. If the effect persists with careful alignment, trying another lens is a good check.

So in most cases: your understanding of a focus plane is correct, and the banded focus you see in macro is usually due to extremely small alignment or distance differences, with lens defects/aberrations being a secondary possibility.