For a vast majority of TC Electronic hardware manufactured in the last 15-20 years (including the popular "Smorgasbord" pedals like the Flashback, Hall of Fame, and HoF), the manufacturing date is actually hidden in plain sight—right next to the serial number.
Specific studio units like the TC 2290 have "Special Numbers" accessible through the front panel (e.g., holding specific buttons during startup) that reveal software revisions and internal hardware versions, though these are technical IDs rather than manufacture dates.
If the serial number starts with 08 or 09 and the electrolytic capacitors are brown, you are looking at a unit from the early 2000s that likely needs a re-cap immediately. Danish TC caps dry out like the Sahara.