The su-exec program in this script is actually the Alpine version created by Natanael Copa.
It is installed from the OP's Dockerfile on line 6.
The Alpine su-exec versionprogram is a substitute for gosu. Both programs are used to enable execution of a command as a specific user with the same environment variables and permissions as would normally be available to the user.
Thus you can effectively "step down" from root inside the container to run a specific command as a user and not as root.
So the alternative solution in Ubuntu would be to install gosu itself, which has no dependencies, and in the docker-entrypoint.sh
script use:
gosu "$ZOO_USER" "$@"
(as per @alexey-shrub's comment below)