It's a real challenge. Any way that anyone, including myself, makes those labels is open to criticism.
One strategy is to use what we did in that paper—a set of scores called the Faris scores that have an array of news sources based upon the ideological orientation of their users. That is one way to measure how liberal, conservative or centrist a source might be.
Another strategy would be to compare the kinds of sources that news organizations quote to the kinds of sources that lawmakers of different political parties quote, and look for correspondence between those. If a newspaper quoted sources that more Liberal parliamentarians refer to in speeches, that might be an example of that paper's being more liberal, and vice versa.
All of these, of course, have their problems. None of them are infallible.