Your Apple ID is the account that you use for everything you do with Apple, like buy an app, shop the iTunes Store, and syncing your device to iCloud. Your old device will still sync photos to iCloud when it's connected to Wi-Fi. As a result, it appears that photos are being uploaded on both phones that are associated with the same Apple ID.
You mentioned that you brought both devices to the store for assistance with transferring your media from the old phone to the new phone. Let's verify if you're using the same Apple ID in both devices by tapping Settings > [your name] > iTunes & App Store. Is the same Apple ID reflected in both devices?