This problem occurs when you are sharing the same iTunes library for syncing. There can only be one copy of an app in an iTunes library. Each app is linked to only one iTunes account. If you both install the same app one of you will have the other's account linked to that app.
Your options are:
1. Live with having both accounts on your device.
2. Choose only one iTunes account for both of you to purchase/download apps.
2. Create separate iTunes libraries and purchase/download apps using your own iTunes accounts and only syncing your device to your own library.