Happy belated birthday, TamaSugarBiscuit!
From my experience, the connection models usually say "happy birthday" (even my Akai in English) at least twice a day. I remember the instructions saying that the tamagotchi may celebrate birthdays making me think that certain stages do not do this. I'd guess those stages are the baby stage and senior stage as I vaguely remember my V3 celebrating mines when it was a toddler and I don't remember if Sotilde, my Hanerutchi in her senior stage, ever did. I know for a fact that the Dream Town will celebrate birthdays throughout the day with a special animation and programmed classical song (it's really nice). I have yet to see my only colour, an ID, celebrate my birthday but I have seen several character birthdays so I'd imagine it is similar to that.
As long as you followed Kuchikid's advice of setting the dates properly you should see some sort of animation, especially since if the tama has that feature it should definitely use it. I'm not sure if there is a way to confirm the originally set data as the only model I remember this happening on was the Dream Town.