In short, Connecticut is not a stop-and-identify state. This means that police would not usually have the right to demand ID from you — that is, they could not arrest or detain you should you fail to produce identifying documents when asked.
Simply asking for ID from another citizen is rarely a violation of rights. By extension, asking for ID is usually within a police officer's authority. In fact, multiple encounters with police would probably show you that it is a relatively common question. Law enforcement agents are trained to gather and record information to use in investigations and prosecutions, and identification cards are one of the most efficient tools in doing so.
Because an official ID gives police a large amount of information about you, it may be to your benefit to politely refuse requests for identification and ask if you might continue on your way, depending on the situation. However, you should probably practice prudence when invoking your constitutional rights. There are many factors that could complicate an incident. Simply because the state does not appear on the Immigrant Legal Resource Center's list of stop-and-ID states does not preclude you from having an encounter with federal agents with potentially different powers, for example.
There are many notable exceptions to the restrictions of police officers' power in to require ID in Connecticut. Some loopholes stem from case law rather than state statute or constitutional law. Some involve police duties and privileges on the scene of a reported crime. Each situation involving law enforcement likely requires care, and, as such, this article should not be taken as legal advice. It is only intended to inform.