When you have the flu you can catch the flu twice or even 3 times in the year but you cannot pass the same flu virus back and forth.
When a flu virus infects your body then your bodies immune system works to fight off and kill that strain of the flu virus.
So after you're no longer contagious with the flu then there's no way to pass that same flu virus back and forth to someone.
However you can get a different strain of the flu virus and then pass that flu from yourself to someone else.
You can also pass the current flu virus you have to someone else while you're still contagious but not back and forth more than once.