Connecticut does have a firearm registry. Older entries are known to contain a number of issues. You can figure some of those out just by looking at the annymized version that was released under FOIL.
Unlike some states (like New York) where all handguns are either registered or contraband (effectively), Connecticut's registry is mostly based on transfer records (i.e. submitted DPS-3 forms that go to DESPP-SLFU and the Buyer's CLEO in addition to the buyer and seller). Those records have been computerized for years. LE in Connecticut have access to information regarding permits and guns associated with an individual. The database can also be used to track a serial number/gun to an individual. Some records on the database also originate from AW Certificates of possession, MG registrations, and potentially voluntary registrations (I am unsure what the process is for that, but the CSL publishes a retention schedule for that information).
Many people will casually refer to the DPS-3 form as a registration form-though technically is a sale/transfer record. (When the AWB was passed, some people erroniously thought that they didn't need to obtain a certificate of possession because they already had a DPS-3. They were wrong.)
As bad as registration is, I personally like the DPS-3 forms. It provides a provenance for where the gun came from, in the event that the origin ever came under question. The reality is that CSP and my town already know that I'm involved with guns due to permit records. They would also already know about the NFA paperwork.
There is no prohibition against a state registry, even a computerized one. Some states have that, but Connecticut does not. (The federal prohibition applies against ATF creating a computer database other than the NFRTR).