with Google cloud platform, cloud SQL you get a lot of options to setup the infrastructure. Does this mean cloud SQL is infrastructure as a service ?
No, the infrastucture of Cloud SQL is managed by Google and by it's engineers, so, Cloud SQL is PAAS (Plaform As A Service).
Cloud SQL is a docker container built on top of a GCE instance, and Google monitor everything for you, and fix the Cloud SQL instance automatically if something goes wrong (Sometimes Google software engineers have to perform some actions to fix some issues if the instance is stuck). So, the only thing that you have to take care of is to store and query your data.
Also, Cloud SQL offers a lot of interesting features, such as, failover replicas, read replicas, user and database adminitration, etc.
So, in Cloud SQL, Google doesn't just sell the infrastucture to create databases, but also the application itself and the monitoring tools too.