My immediate thought, with no further analysis, was that the point made in the video (that trade would lead to less diversity inter-culturally, and more intra-culturally) may be likened to factor prices and specialization in the Factor-Price-Equalization Theorem (some folks attribute FPE to Paul Samuelson, others to Bertil Ohlin - while it seems to be generally accepted that Abba Lerner developed it earlier, independently of the other two).

But it doesn't add up intra-culturally. In FPE the factor prices will move towards each other in each country - the dearer becoming cheaper and vice versa (and "double vice versa" in the other country). With the culture, diversity is increased within each country.

Still, I think, that the video may illustrate a demand-side version of FPE - that tastes and preferences will become less diverse, or diverging, inter-culturally as a result of trade (the respective indifference curves of the two countries will move towards each other). This development should lead to more trade and more factor-price equalization, solely on account of the changes in demand (adding to the supply-side effects of FPE).