The init containers do indeed check for database availability. You can check
kubectl logs kong-kong-init-migrations-87r74 -c wait-for-postgres
kubectl logs kong-kong-6447659bbf-9jl9d -c wait-for-db
Those should roughly indicate what’s failing when they try to connect. The first should generally fail with either a DNS failure or some sort of TCP connection failure. The latter seems more likely; I’d first check to confirm that you have the necessary AWS network ACLs and security restrictions to allow traffic between your EKS workers and the database instance:
https://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/USER_VPC.Scenarios.html
https://docs.aws.amazon.com/eks/latest/userguide/sec-group-reqs.html