哦，抱歉，樓主既然用全英文提問，客隨尊便，我也改用英文繼續回答吧。
The Third Anniversary of a marriage is an extremely important date in the western world and in western culture. The first anniversary is Paper Wedding, followed by the second equivalence of Cotton Wedding. The Third Anniversary is traditionally called Leather Wedding. From paper to cotton then to leather, the relationship becomes really “consolidated” upon the arrival of leather. Paper or cotton can be easily torn apart, but not when it becomes leather.
Leather brings in real essence to true love and marriage. As a result of such, additional value has been imposed on the third anniversary of any marriage. In traditional European countries, lovers (if they are not married) also celebrate the third anniversary of the date they first met. In other word, leather is a milestone for a marriage.
Some people believe that if a relationship can come across the third anniversary, it will last forever. Of course, you know and I know that this is not likely to be true for many married couples nowadays.
Due to the importance of the third anniversary, many prefer to call their sixth anniversary the Second Third Anniversary.
Therefore, “this is our Second Third Anniversary” means the sixth year of the wedlock. It does not mean 23 years, it means 6 and not otherwise.
As stated in the first paragraph, I really have no idea as to what movie you watched. However, the specific scene you witnessed should be a wedding party, or a candlelight dinner, or whatever but definitely happened on the day of the sixth anniversary. If the couple were surrounded by friends and relatives, someone might even say “We look forward to your third Third Anniversary”. The expression of Third Third Anniversary DOES exist in practical English.
I hope my answer clears all confusions.