The mainly difference, which help to decide the selecting decision for the software design is the embedded mode. H2 supports this kind of the mode while PostgreSQL doesn't support it. Further H2 is platform independent and mostly used for the small application and PostgreSQL only supports some kind of the OS.
Serving multiple websites on a single host with Docker is against the principles of Docker and micro service architecture. Sure you can do it but what is blocking you from separating it into multiple docker instances. That way, if there is a problem not all of the sites go down. By separating the hosts, you make it easier the diagnose possible problems and minimize down time.
To meet this requirement it is necessary to define proper order of tabulation by setting "tabindex" attribute and also provide the possibility to work with elements (set the checkbox, change value of radio button, submit the button, etc.)