There are a few elements to this answer, so I'll try to tackle them one by one.
is it possible to brew another cup with the used grounds...?
Yes. It is physically possible to do this. Just do what you described and voila, more liquid into the next mug.
this would be the same as using more water and brewing over a carafe to brew twice as much coffee
This is where you are wrong. Brewing coffee is chemistry, whether you are using a Chemex or an espresso machine or a cheap drip machine. Each stage of brewing produces different flavours from the coffee. I can't guarantee this is accurate but here is an example of what I'm talking about.
The first part of your pour controls the bitterness of the coffee. The bloom helps to get rid of CO2 and dissolves oils on the beans and in the grounds. It will develop your "body".
The second part of your pour will determine the sweetness. Some people swear by small pours over a longer period of time, while others prefer the ease of just slowly filling the brewer evenly.
Finally, the last half of your pour controls the intensity of your cup. This is where your question comes in. More water in this last section will produce a weaker cup, and less water a stronger one.
All of these parts of the pour need to be melded together to create a delicious cup of coffee.
Pouring your first cup then re-using the grounds for the second one means the second cup misses out of the main part of the brew. Yes, the liquid will tastes like coffee, but will not have very much caffeine, and will lack the flavours present in a well-brewed cup. You're basically making ultra-weak, bland coffee.
If you want to make more coffee and don't care about the strength, brew a larger batch into one carafe, then swirl and split it between tow cups. You should adjust your grounds-to-water ratio accordingly though.
I hope this helps you answer your question!