I can confirm that installing bash-completion enables some completion, but does indeed seem to not complete for package names. I guess it'd probably be interesting to try reproducing in a more "vanilla" chroot and see if it's caused by something we've done or whether it's something that needs to be reported upstream:

FROM ubuntu:yakkety

RUN apt-get update -qq && apt-get install -yqq bash-completion && rm -rf /var/lib/apt/lists/*

$ docker build -
...
Sending build context to Docker daemon 2.048 kB
Step 1/2 : FROM ubuntu:yakkety
 ---> 31005225a745
Step 2/2 : RUN apt-get update -qq && apt-get install -yqq bash-completion && rm -rf /var/lib/apt/lists/*
 ---> Running in f2652cdfb4fd
debconf: delaying package configuration, since apt-utils is not installed
Selecting previously unselected package bash-completion.
(Reading database ... 6513 files and directories currently installed.)
Preparing to unpack .../bash-completion_1%3a2.1-4.3ubuntu1_all.deb ...
Unpacking bash-completion (1:2.1-4.3ubuntu1) ...
Setting up bash-completion (1:2.1-4.3ubuntu1) ...
 ---> 3fbf18528d4e
Removing intermediate container f2652cdfb4fd
Successfully built 3fbf18528d4e
$ docker run -it --rm 3fbf18528d4e bash --login -i
root@c65f6c083a4c:/# apt-get update -qq
root@c65f6c083a4c:/# apt-get in^I # completes to "install"
root@c65f6c083a4c:/# apt-get install libopen^I # (nothing)

Upate: I can confirm that issue for a quick solution I use ZSH, there the bash_completition is fully working, but not with bash.

I run into this today. (I'm using docker to run ubuntu 1604 opengl GUI software in ubuntu 1804)
I think I figure out why this happens and how to "fix" it for my usage

There are two problems here:

1. bash-completion is not installed
   $apt-get install bash-completion
2. the cache file used by apt completion is deleted by docker apt setting
   $ rm /etc/apt/apt.conf.d/docker-clean
   After this, restart the bash and run a apt-get update should enable the bash-completion

Long explains of the issue:
The bash completion use a command to generate the package completion list:

/usr/share/bash-completion/completions/apt:190
COMPREPLY=( $( apt-cache --no-generate pkgnames "$cur" \..

however, this apt-cache --no-generate pkgnames $cur has problem in the docker version

$apt-cache --no-generate pkgnames test
E: Could not open file  - open (2: No such file or directory)

This is because of the docker-clean apt setting.

#/etc/apt/apt.conf.d/docker-clean
DPkg::Post-Invoke { "rm -f /var/cache/apt/archives/*.deb /var/cache/apt/archives/partial/*.deb /var/cache/apt/*.bin || true"; };
APT::Update::Post-Invoke { "rm -f /var/cache/apt/archives/*.deb /var/cache/apt/archives/partial/*.deb /var/cache/apt/*.bin || true"; };
Dir::Cache::pkgcache ""; Dir::Cache::srcpkgcache "";

Two problems for the completion:

1. Dir::Cache::pkgcache "" which is the reason for the emply filename of the error message open file - open (the filename supposed to be before the -)
2. the setting will try to remove the pkgcache file after each apt invoke

I can understand why we want to remove the pkgcache. So my solution is just a partial fix. But it serve my purpose well. My docker is a long run container and I prefer the cache.

Nice, good find! 👍 (and thanks for giving such a detailed explanation)

Closing, since this is caused by configuration that's by design (and the workaround is now documented here, and for me is the second Google result for "docker ubuntu apt-get bash completion").