A plane is just a polygon that (typically) covers most, or all, of the PCB area. Generally when people talk about a ground or power plane they mean a dedicated layer, so your example images would more correctly be called ground (or Vcc) fill. This is commonly done on 2 layer PCBs to get some of the benefits of a dedicated plane without the extra cost of a multi-layer PCB.
If you have a polygon creating ground or power fill on the same layer as other traces, then it will be broken up by the traces - as show in your images. This can mean that you get islands (Eagle calls them orphans) that are not connected to the rest of the net. A setting on the polygon determines if Eagle will hide or show those orphans. Having orphans on can be misleading, but DRC should warn you about it. Planes with their own dedicated layer will only be broken up by vias, holes, slots, etc - but on a dense board that plane can look like swiss cheese. Via stitching is often used between layers to reduce impedance and connect any islands.
I was only aware of having GND plane in a separate layer and thought each plane has its own layer. Can we have GND and Vcc planes in the same layer?
You can have multiple polygons on the same layer connected to different nets (e.g. GND and Vcc), but obviously they cannot overlap. Look at more sample boards that have been published to get a feeling for how these are used - generally for power distribution or thermal control.
Note that in Eagle you should use different "rank" settings on polygons on the same layer to make sure they don't overlap. Rank 1 is the most important, rank 2 less important, etc. If you forget this and the polygons create a short then it will be highlighted by DRC.