Where does Apple App Store download security updates to?

First off: did you consider "Content Caching" in High Sierra's System Preferences > Sharing?

But to answer your question: while downloading you'll usually find the files in the system caches in the...

/private/var/folders/zz/

...folders (owned by _softwareupdate or/and _nsurlsession, with no access to the user). But that's just a temporary state and they usually get assembled in numbered directories into...

/Library/Updates/

Since updates consist of more packages these days, they refuse to run individually and the .dist file will have to be opened with the Installer.app to install them in order.

The packages will wait for installation, before being removed, so making sure System Preferences > App Store is set to not automatically update and clicking "Not Now" before the download, or "Later" in the installation window will postpone installation and therefore removal.

If you need to update multiple Macs then you may be better off to bypass the Mac App Store app and download updates manually. The safest way to do that is directly from Apple.

To do this, use your browser to go to: https://support.apple.com/downloads

Once there you'll see the most recent software updates available for download. However, you can also use the icons at top of the page to browse downloads by product.

Using the example in your question, you can click on the macOS icon to filter the list by macOS downloads. Doing this will provide you with access to Security updates, combo updates, etc relating to macOS.

You also have the option of using the Search field to search for a specific download. For example, entering Security update as a search term will present you with Security Updates going back to 2003. You can also search for specific updates relating to OS versions or Apple hardware.

Typically downloads will be in the .dmg disk image format and you can then keep and use these as required.

Finally, if you're going to manually download updates, you may want to consider reviewing you settings in Apple > System Preferences > App Store.