I’m surprised no one has mentioned AWS ElasticBeanstalk. It supports Docker (ECS behind the scenes) and is very close to Heroku in terms functionality but with EC2 pricing.
You can launch a DB through EB but the docs basically state it's a bad idea (as it gets taken down when you delete your app, which you might find yourself doing if EB gets in to an unrecoverable state).
So now you have to manage EB + RDS separately, which should be automated so now you need CloudFormation to add the security groups and manage the vars for the DB connection properties in your EB.
Beanstalk is nice for rapid development. The problem I is that it doesn't scale for nontrivial architiched apps. I've converted failing beanstalk deployments to real orchestarators a few times.
You can launch a DB through EB but the docs basically state it's a bad idea (as it gets taken down when you delete your app, which you might find yourself doing if EB gets in to an unrecoverable state).
So now you have to manage EB + RDS separately, which should be automated so now you need CloudFormation to add the security groups and manage the vars for the DB connection properties in your EB.