If you boot your Mac from a backup volume, CCC will be started upon login to ask whether you'd like help restoring from that backup volume. Sometimes, though, this offer is made when you're booted from a production volume, not a backup. CCC makes this assessment based on your currently-defined backup tasks. If you used CCC to migrate from one drive to another, then the task that you used to perform that clone will still be present on your new startup disk. When you boot your Mac from the new disk, CCC will see that you have a suspended task that specifies the current startup disk as the destination, thus giving the appearance that your Mac is booted from a backup.
If you migrated to a new disk and you'd like to avoid CCC opening on startup and offering restore guidance, open CCC and delete the task that you used to clone to your current startup disk.