Whether you get more points or not depends on the ‘Preserve other end’ setting in MatchSrf and/or the Refine setting. You don’t say what what you rebuild to but for tangency all around degree three, and four points in each directiion would be the starting point. If Refine is on in the MatchSrf dialog, you may well get more points.
Position matching uses the row of control points along the edge.
Tangency matching uses the two rows of control points along the edge.
Curvature matching uses the three rows of control points along the edge.
Minimum number of rows of control points = Number needed for desired continuity on first edge + number needed for desired continuity on second edge
If needed MatchSrf increases number of rows control points by increasing degree of the surface to achieve desired continuity on the edges. If the number of rows of control points is adequate the degree of the surface is not changed.
The Refine option in MatchSrf increases the number of control points along the rows so that the requested degrees of continuity is achieved within the tolerances along the edges.
Rebuild only matches position along edges at a number of locations equal to the number of control points along the edge. Any preservation of tangency is coincidental.
MatchSrf changes a single span degree 4 to degree 5 if the matching condition is curvature and the preserve condition is curvature. Curvature requires 3 rows of control points so 3 + 3 = 6 rows of control points are needed, hence the change to degree 5. For any other combination of match condition and preserve condition 5 or fewer rows of control points are needed so a single span degree 4 is sufficient.