One potential bias is, say, that a data set of loans shows that in a certain city, people born in one neighborhood are more likely to pay back large loans without defaulting than people born in another neighborhood. An algorithm could, based on this data, conclude that it should not issue large loans to anyone from the second neighborhood, because it lacks data that those people are usually but not always poorer, and thereby encode bias based on place of birth.
But the paper is talking about a different kind of bias, say, that a data set of loans has thousands of data points of people born in the first neighborhood and tens of data points of people born in the second neighborhood. Even if you were to control for economic status (or perhaps explicitly control for things like ethnicity), an algorithm that performs well on the data set as a whole might perform poorly on the particular subset of people born in the second neighborhood, simply because it has less data. The algorithm might have an acceptable (to its human supervisors) false-positive rate rejecting loans to people born in the first neighborhood, but the exact same model might have a much higher false-positive rate to people born in the second neighborhood.
That's different, and that effect could apply even if people in the second neighborhood were just as good as paying back loans, because you have fewer items in your sample and so there's more noise. That's what the section "Measuring complex trade-offs" is about.