What exactly does Ubuntu share with other Debian based distributions? What do all Debian based distributions share in common that other distributions don't?
All Debian-based distributions share the Debian package archive either directly or (in the case of Ubuntu) by making their own rebuilds of it. Thanks to this a Debian-based distribution instantly has effectively every piece of software you'll ever need available in a package. We also get the upgrades and bug fixes for these packages made by the 1000 or so Debian Developers.
We also all share the dpkg and apt package management tools. Dpkg provides the most reliable package install/uninstall/upgrade as it is transactional. If a package has an issue installing the install is rolled back completely, you don't get something half-installed. Apt provides powerful dependency resolution and allows us to provide three levels of dependencies to ensure users get the best experience from the package but are still able to make some customizations to the packages installed by choosing to install or not a packages Recommends and Suggests lists.
Once you get below the general look and feel differences you may notice some of the best aspects of Ubuntu are actually from Debian. This is by design, we are deliberately trying to take those advantages and build on them. All other Debian-based distributions will have those same advantages.