As a relativist myself, I'd say that this description misrepresents Relativism, out of a misunderstanding. It's not that there are no facts to discover, but that whatever you discover is a mental construct that is mediated by how you think about it. This "nihilism" is popular because it's rationally sound.
"Whether or not there is such a thing as a chair is simply a matter of convention", true, but that's because _being_ is not an essential property of the universe that you can get to know, but a metaphysical concept that you think about; as such, the facts you get to assert about it are dependent on how you define the concept. As different schools of philosophy have their own ways to think about the concept, they'll arrive at different conclusions about its characteristics.
In the end, all philosophical enquiry is a mental exercise that produces knowledge in the minds of us fellow humans, and it's mediated by the capabilities and limitations of our brains.
In the same way, all rational discourse takes the form of discovering new facts that are derived from some stated axioms and following defined rules of interference. There are no metaphysical axioms and rules of interference existing outside our brains, so rational thought necessarily depends on the shape of -and the consequences implied by- the particular principles you chose to use. Choose different ones, and the shape of your reasoning will vary (even though, as you said, it is often possible to translate between different initial perspectives).