The Mac App Store (MAS) registers downloaded apps—paid and free—to the Apple ID associated with the store at the time of download. Most people only ever log in with a single Apple ID for app purchases or retrievals, and thus all apps installed on their Macs remain correctly licensed to that Apple ID.
However, it’s possible even using a single Apple ID that you will see an error when you attempt to update certain apps that reads, “Update Unavailable with This Apple ID.” This seems to happen commonly with Apple’s productivity apps: Pages, Numbers, and Keynote.
The message goes on to say, “This update is not available for this Apple ID either because it was bought by a different user or the item was refunded or canceled.” If none of those conditions are true, you may still see the error in some readers’ (and my family’s) experience. (Why does this happen? Apple hasn’t provided any insight.)