The answer is no, when you stop your container it is not deleted, only stopped, you can start it when it is stopped the same when you stop your computer it will not vanish from your desk :)
You can even restart it when it is running, same as you would do with your computer.
However when you remove/delete the container with
docker rm -f containername
or
docker-compose rm
then it is truly deleted, equivalent of making your computer vanish from your desk.
But even than you can still persist your data with volume mounts, for example in your compose file your ./database directory will not be deleted from your host machine even when you delete the containers using it. It is the equivalent of using an external usb drive in your computer, so when you make your computer vanish from your desk with deleting it, you still have your usb drive with the data on it that was there when you still had your computer.
So you can persist your database files with the same technique in a volume mount like this:
services:
postgres:
image: postgres:11.6
env_file:
- .local.env
volumes:
- ./database/:/docker-entrypoint-initdb.d
- ./postgres-data/data:/var/lib/postgresql/data
ports:
- 5432:5432
...
This way when you delete your container(s) and do "docker-compose up" again for the same compose file, postgres will not run its init scirpt because the /var/lib/postgresql directory is already populated in it.
However, my computer analogy is valid only in this context, please do not think of containers being mini computers or mini virtual machines, they are not! But that's an other discussion.