Really? The occasional oversized telegraph pylon or well-restored aside, the most notable architectural styles in both countries tend to be Norman-era castles and Gothic religious architecture, both of which have essentially French origins, and the occasional Roman ruin. Both countries have majority populations who are ethnically Germanic.
As neither of these is very arguable, that leaves culture, which I've argued before it's hard to see as very distinct on the scale of inter-civ differences we see when considering the scope of civs in the game. 80% of words in the English language are of French origin, and many of the remainder are shared due to a common Latin or Greek root (in contrast the Khmer and Thais don't share a common alphabet and their languages have different roots); the major linguistic differences are in grammar. French was the national language in England throughout the Norman period (fair enough, you can point out that the Thais spoke Khmer during the Khmer period, but that was during the period of Khmer rule. French was the national language of an independent England for over two centuries).
Both have similar political outlooks, histories shaped by similar struggles for dominance on the European stage (the main difference being that France has been more successful), both were major players in the age of European colonialism (the main difference being that the British were more successful).
What we recognise today as cultural differences between the countries - such as foods (which aren't very distinct any longer as French cuisine is a significant component of internationalised Western cuisine) and, someone mentioned on another thread, family traditions - aren't things that are 'visible' on the scales Civ games deal with.
Personally, I've found the modern French to be more culturally familiar as a Briton than I find the Americans (although admittedly that may be due partly to long exposure and due partly to having lived in the US and become more aware of the differences).
With regard to the languages, while there are indeed a good number of shared words (though nowhere near the 80% mark - estimates of French-derived English words are usually closer to 30%), the same can be said of Thai and Khmer, which also have a long history of linguistic cross-pollination. However, like those two languages, English and French belong to fundamentally different linguistic groups, with English being a Germanic language and French a Romance language.
And as for other aspects of culture, there will always be a good number of similarities between neighbours, but there are also a huge number of divergences, from the political organisation (England has a strong tradition of strong local government, compared to France's history of centralisation) to the economic focus (where somewhat paradoxically the English system is disproportionately centred on the capital), to the general outlook (with England's maritime prowess differing sharply with France's continental - their respective successes imperially and on the continent are not coincidental).
Anyway, as I said originally, I agree with your main point that there is a problematic tendency to downplay the differences of cultures that are different form our own. But it's also possible to go too far in the other direction and place undue importance on every divergence while ignoring similar patterns closer to home.