I could not find anything like what I'm thinking, though it is hard to search for 'snapping' without getting a lot of mixed results.
What I'm looking for is an add-on or behavior where the vertices of your mesh form an implicit 3D grid system, so that you can snap to the closest grid planes automatically.
Example: Given a usual triangle, say you want to move the bottom right vertex to the top of the triangle (only z). You move that corner vertex upwards and once it aligns with the top vertex in the z-axis, it would snap to that z-value.
Most of the snapping I've seen is to the vertex/edge/face itself, this is more snap to align with closest.