A new EC2 instance was created and the content is going to be migrated to it from the existing instance with Plesk.
Is it safe to switch elastic IP to the new instance after migration? Will it cause any issues?
Such operation should not cause any issues. Plesk Migrator will migrate all sites to the private IP address of the new instance.
After migration is finished, just associate the elastic IP with the new instance in AWS Console.
Also, in Plesk set the Elastic IP as a public IP address for the private one in Tools & Settings > IP Addresses > 192.0.2.2 > Public IP address.