Shazam explains that if you don't want your data to be automatically transferred, you can delete your Shazam account within the app. This will erase all personal data associated with Shazam, including your email address and connected Facebook account. Shazams will remain in the app but they will no longer sync between devices and they will be gone if you delete and reinstall Shazam.
The company points out that analytics logs and how you used Shazam won't be erased even if you delete your account, but the logs will no longer be personally identifiable at that time. If you never created a Shazam account and continue to use Shazam under Apple's ownership, any Shazams recorded will not be associated with you.
At the time of the acquisition announcement, Apple said that it was "thrilled" for the Shazam team to be joining Apple. Down the line, the company hinted that users can expect more integration with Apple Music and Shazam: "Apple Music and Shazam are a natural fit, sharing a passion for music discovery and delivering great music experiences to our users. We have exciting plans in store, and we look forward to combining with Shazam upon approval of today's agreement."