Hi Emily,
Diarium does not directly “connect” your Mac to your iPhone, and there is no Diarium account.
The message means that the App Store found no Diarium Pro purchase for iOS. macOS and iOS are separate platforms and require separate purchases, even when both devices use the same Apple account. Seeing your macOS purchase in the purchase history on your iPhone does not make it an iOS purchase.
This is stated directly on the Pro Version purchase page and explained in the Pro Version FAQ.
To use automatic sync, Pro must be active on both platforms. You can then select the same cloud-sync provider and account in Diarium on your Mac and iPhone.