I don't think failing the build would be correct for everyone, I would propose treating Error status as a failing test and mark the build as unstable as is done for failing tests.

I guess that would be ok for us as well, we have configured the build job to fail on unstable anyway.
 

Would it be possible for you to enable the option to save the JUnit files that are generated and then upload the resulting JUnit file? The current XSLT transform should actually work for the case of setting it to unstable, so I want to see what the resulting transform creates.

We could do this if you can tell me how. This might look like a stupid questions but I have no idea how to enable keeping JUnit files. There is no option for this, so what should I do?

The option is in the global configuration area for NUnit.

Manage Jenkins > Global Configuration

I looked through all the "Manage Jenkins" but I couldn't find anything related to nunit settings. There is no "Global Configuration" in "Manage Jenkins" on our server.
 

It looks like debug mode is only available if you connect via a debugger, so that doesn't help much. I'll just try running your file through the xslt transform locally.

So, what looks like is happening is the NUnit XML output supports a failure at the test-suite level, which I am not sure the JUnit format supports (which is used to display the results). The schema for the JUnit format, and what is supported by the JUnit plugin is not very well documented, so I will need to determine if its possible to support this sort of thing.