After several days of digging through the Ruby sources and testing I figured out what went wrong:

Testing the repo in subversion.rb

After some reading I modified some lines for using another method in subversion.rb:

        ##
        # Checks the status of this repository and throws unless it can be accessed
        # correctly by the adapter.
        #
        # @raise [ScmUnavailable] raised when repository is unavailable.
        def check_availability!
          # Check whether we can access svn repository uuid
          cmd = build_svn_cmd(['info', '--xml', target])
          xml_capture(cmd, force_encoding: true) do |doc|
          #popen3(['info', '--xml', target]) do |stdout, stderr|
          #  doc = Nokogiri::XML(stdout.read)
...

Now I am getting something in the logs:

App 336 stdout: SCM command failed: Non-zero exit code (pid 368 SIGSEGV (signal 11) (core dumped)) for `/usr/bin/svn`
App 336 stdout: Error output is svn: warning: W000013: Can't open file '/root/.subversion/servers': Permission denied

That’s more helpful than nothing.

Incomplete change to user “app” in Docker container

With the error message above I could analyze the problem further.
The Passenger processes may run as user “app” but their environment still points to that one of “root”. After trying some configuration method, which didn’t work, I just modify the “web” script in

/usr/src/app/docker

by adding following two lines

#!/bin/bash -e

export HOME=/home/app
export USER=app

BIND="${BIND:=0.0.0.0}"
PORT="${PORT:=8080}"
...

Conclusion

With the two fixes above I could enable the SVN access successfully. Maybe this will help others in the future too.