Will the Premium subscription work across multiple devices?
The subscription to the Premium Account is connected to Apple ID (for iOS) / Google Account (for Android). The Premium can be used on several devices in case of being logged in to the same Apple ID / Google Account for which the purchase of the Premium was made.
For iOS applications you can configure Family Sharing for several Apple IDs: How to use Family Sharing in iOS applications.
The synchronization of business cards in Business Card Reader app does not require the Premium subscription and can be performed in the free version: How do I synchronize my business cards?
The user will see the synchronized business cards in BCR but will not be able to add new cards and use the Premium features.