It's always best to avoid any discussion with law enforcement, even if you have nothing to hide.
The Supreme Court in Hiibel v. Sixth Judicial District Court of Nevada, 542 U.S. 177 (2004) upheld a Nevada statue which required anyone detained during a Terry Stop to comply with a demand for identification. Colorado has a statute similar to Nevada's which allows an officer to "request" identification rather than compel. However a failure to produce identification most likely would not deny the basis of the officer's reasonable suspicion may cause the individual detained to run afoul of other laws like obstruction. This is an unsettled area of law in many states so there is no hard and fast rule for what happens when a person refuses to identify themselves nationally.
Nevertheless in this case the officer seems to have specifically detained the author under the reasonable suspicion that he solicited an illegal vehicle for hire. Once the passenger told him that he was in an Uber cab (a lawfully hired vehicle in Colorado) the basis for that suspicion was denied and the author should have no longer have been detained or asked for personal identification.