Yes, vertex color in Blender is a "face-corner" attribute: you can specify a different value for each corner of a face:
The trick here is to enable "face masking mode" with the button immediately to the right of the "vertex paint" mode select. When you do this, you can select individual faces (either with vertex paint's select tool, or via face selection in edit mode) and your painting will be restricted to corners that are a part of that face.
There are drawbacks to doing so-- just as with other vertex attributes, many engines will need to split any vertices to limit vertex color to a face-corner. I can't say for certain whether Crash Bandicoot used per-corner vertex color like this. Another way to do it, that isn't actually any more expensive in most rendering engines, is simply to rip the edges of discontinuous color, so that you have duplicated vertices/edges located in the same positions.
Thanks for mentioning in comments about the new, different types of vertex color layer attributes: Blender can now support both vertex-data and face-corner-data vertex color layers. You choose upon creation of the vertex color layer. If you want a different color for each corner, you need to be painting on the face-corner type.