Not exactly the same. As I mentioned in the comment, you have direct access to the Linux OS when you host. Open as in access to the underlying OS is open. You can configure it, install packages that you want, etc.

When you run AWS Elasticsearch, you have no access to ES instance. You can't configure it, you can't add your own plugins. You are no longer a user, you are just a consumer of the UI/API, even when the primary product is still called Elasticsearch.

Big difference.