Do the same rules apply to repointing stone?
In general terms, yes. Stone buildings with a majority of the pointing intact can be patch pointed. Mortar can be made compatible with the existing in composition, color, and joint style. As with brick pointing, proper joint preparation is critical to a lasting job. All of the original pointing mortar should be removed down to the softer bedding mortar. High portland cement mortars can be problematic when used on stone. While most stone, such as common field stone, are quite hard, some are not. Just as soft brick may be damaged by dense mortars, so too may softer stones, such as limestone or sandstone.
