The guy interviewed is himself a liberal.
He's gone into this numerous times. FIRE's stated rules are that they consider:
1) State institutions who are legally required to allow free speech.
2) Private institutions that promise free speech.
They take the position that any private university which states up front that it enforces a code of conduct that takes precedence over free speech (such as religious rules), is not violating the law, and is lying to students about what they should expect. They don't support the limitations these schools place on free speech, but as private institutions they feel it is their right to hold them.
You may disagree with that philosophy, but there is nothing inherently liberal or conservative about it. It's a defensible response to living in a pluralistic society.
Now, it so happens that the majority of speech restrictions from public schools at this time in history are based on liberal concepts. Conservatives tend to retreat to private religious institutions for their discrimination. But FIRE *has* gone after traditionally conservative/catholic institutions who promise free speech and then renege on it.