Good detective work! In essence, users are not automagically granted access to sync to their cloud-based Dropbox folder. They can install and use the Dropbox app to establish sync permissions at the OS level. Or they can use a method built-in to an app to grant the app access to send its content the Dropbox cloud folder.
Both methods require a verification step. Once the verification is established, the key/security signature that is required to log-in is stored separately from the Dropbox app and its local/cloud folder. Hence, even when the Dropbox app is removed, one may still be able to access the Dropbox cloud folder through an app using its own mechanisms that bypass the need to have the Dropbox app installed. Finally, if the key/security signature to access Dropbox is removed from the device in some manner, log-in permission to Dropbox has to be manually re-validated.
FWIW, with iCloud in the Apple ecosystem, validation to share using iCloud is established by being logged in to your AppleID account on the device at hand.
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JJW