Your are perhaps confusing 'vertex' and 'point'.
An edge has a start vertex and an end vertex.
This vertex can belong to more edges depending on the 3d geometry, and context arrangements.
A vertex has a 'position' which is represented by a 'point' in space.

In newer SketchUp versions an Entity has a persistent_id.
A 'vertex' is an Entity.

If you move a vertex in space its 'position', and therefore its 'point', can change, but its persistent_id remains intact.

If you have two edges sharing a vertex, then that vertex is a single entity, and has one persistent_id.
But if you subsequently separate the two edges into different contexts - e.g. one is now moved inside a group - those two edges have been changed and each now has a separate vertex where they shared one, the vertex that stays in the original context might be expected to keep the persistent_id, but the one that is now inside the group's context is in effect a new vertex and gets its own unique persistent_id.

If SketchUp did not do this, then chaos would result with persistent_id's !

You can track your edges by persistent_id as these move en mass into any new context - with their vertices can get recreated when shared ones get split-off in this way.
Use a persistent_id for each edge and reference its start/end vertices etc...
The edge will keep its persistent_id, but some vertices might not...