With starling bank, you have separate authentication for their personal and business accounts. This means that unfortunately you can only have one or the other actively syncing to Coconut at a time. You can handle this one of two ways:
Connect your Starling business account to Coconut and not your personal account as you won’t be able to keep them both synced at the same time. If you have transactions on your personal account you wish to add you can add them as manual transactions.
Connect both accounts to Coconut initially by going through the connection flow with your personal account number, and then re-syncing the connection using your business account number. Every now and then (eg monthly) swap the active connection by going to More > Connections and by going through the re-sync flow for Starling with the relevant account number. This will sync the missing transactions since it was last active. Then swap it back to the other account the same way.