That is most definitely NOT the way Aria should work. As a matter of fact, I'm not even sure how that could possibly occur.
I tried to upload some shots of a normally-functioning Aria Player, but it won't let me for some reason. If you click on an instrument slot you will see your options in the drop-down menu. In your case, you may prefer to select the Notation instruments.
If you continue adding different instruments they will load in their chosen slots and will all remain in the same Aria window.
Even if you should choose an Ensemble (top left) you will see all the chosen ensemble's instruments load in their assigned slots, all in one Aria Player window.
If you are unable to get Aria to function properly, the first thing I would do if I were you is reset it by going to your user>Library>Preferences and deleting com.Garritan.ARIA Player.plist and com.Garritan.ARIA PlayerSA.plist. New .plist files will be created the next time you launch Aria Player. If you're nervous about this you could always just drag them to your desktop and re-launch Aria Player . . . you can move them back if necessary.