Vertex paint is not the same as texture painting.
Vertex painting is painting color values directly onto the vertices themselves.
Texture painting is painting an image that is mapped onto a model via UV coordinates.
I think you want to be texture painting.
Switch to the texture paint workspace, In the texture editor window, add a new texture at the resolution you want. Make sure that image is assigned to the mesh in a material. Then you can paint in both the 3D View and the texture view.
If you really want to use the vertex colors as the mesh material colors and bypass texturing altogether then you need to use an attribute node and set the Base Color value using that vertex color.