Here's an important tip about volume sharing in docker:
Multiple containers can also share one or more data volumes. However, multiple containers writing to a single shared volume can cause data corruption. Make sure your applications are designed to write to shared data stores.
In this context, does Postgres designed to write to shared data stores?
In other words, is it safe to run multiple Postgres containers (possibly with different minor versions) working with same database files located at the data volume?
You cannot have multiple PostgreSQL installations run against the same shared data files, this is a sure recipe for data corruption.
If your need is to update PostgreSQL without downtime, you'll need to use a replication solution that works between different major PostgreSQL versions so that you can first build a copy of the database with the new version an then switch over quickly in a controlled fashion. This still causes a small outage that has to be handled by the application.
Replication solutions that can be used are external replication tools like Slony-I or logical replication. Logical replication is fairly new, it will ship with PostgreSQL v10 (which won't help you with a current upgrade problem), but you can use it with pglogical from PostgreSQL 9.4 on.