Across the history of colonization and colonialism, I always find it such a mystery how some colonies permanently retain a cultural connection to their founder, and others do not.
As the British and later Americans expanded territory across the continent, they obtained lands from a variety of different pre-existing colonies: Florida from Spain, Delaware from Sweden, New York from the Netherlands, Alaska from Russia, etc. And in every single one of these cases I listed, any iota of influence from the original colonizers have long since melted into the greater Anglo-American population, such that we rarely ever think of these states as being anything other than simply "American" and not Spanish/Swedish/Dutch/Russian/etc.
But the British annexation of Quebec after the Seven Years War seems to inexplicitly break that pattern. Even though France had only colonized the region for less than 170 years (not much more than the colonies I listed above), they somehow became so permanently fixed that Quebec has stubbornly and persistently held on to their French heritage for the last 250 years since. How did the French legacy in Canada become so much more permanent than any other colony annexed by Britain?
There's always more to be said, but in the meantime you may find some answers in this thread by u/enygma9753 about the political climate in the British province of Quebec and the Maritime colonies during the interim period between the fall of New France in 1759-60 and the start of the American Revolution in 1775.
James Murray, who had served under General Wolfe at the Plains of Abraham and the capture of Quebec, became military governor of Quebec in October 1760, and then the civil governor of Quebec in 1763 at the end of the Seven Years War. He recognized that, as new masters of a French-Catholic population of some 80,000 (that outnumbered the British 25:1), British colonial officials needed to be mindful not to incite rebellion among their new subjects in Quebec. The arrival of British settlers in the aftermath of the Conquest would cause tensions that threatened to disrupt the uneasy peace with the locals.
British merchants assumed that they would automatically benefit from the anti-Catholic practices codified in British laws and expected the preferential treatment of Protestants, as they had throughout the Empire. Catholics were already barred from the highest offices in the colony and could not vote or run for public office. They had expected to displace the local gentry and recreate a form of Protestant Ascendancy in Quebec -- similar to that in places like Ireland -- which would dominate the political and commercial interests of Quebec.
Instead, Murray was sympathetic to his French subjects and permitted the continuation of their civil code, property rights, as well as easing restrictions on their faith and language. French-Catholics would also be allowed to serve in minor local offices. This alienated the British merchant class, but these civil liberties granted to the Quebecois would be confirmed into law in 1774 with the Quebec Act -- one of the Intolerable Acts that triggered the American Revolution. The Act had also ceded to Quebec a large swath of territory west of the Appalachians to the Ohio Valley, enraging American colonists who sought unlimited westward expansion.
When Guy Carleton became Governor of Quebec, he continued this policy of accommodation and tolerance. While he was naive in assuming that this policy earned him the devotion of the Quebecois people, it ensured that the bulk of the population would not raise arms against the Crown when hostilities broke out in the Thirteen Colonies. French-Canadians lived in peace under British rule and the Patriots' vague offers of liberty did not hold the same weight as the rights guaranteed to Quebec under British law. While the French-Canadians were still leery of the British, they trusted the Americans (who were majority Protestant and largely anti-Catholic) even less and were deeply suspicious of them. The accommodation of French language rights continued throughout British rule in Canada and were reconfirmed in 1867 with Canadian Confederation.
In America, both federal and state laws in the 19th and early 20th-century pushed for the assimilation and anglicization of non-anglophone citizens. English was to be the language of instruction in schools and the official language of the government and business -- even in states like Louisiana with a large French population. The French language would be suppressed, or diluted with competing dialects like Spanish and Creole. In states where French settlement was limited, overwhelming English-speaking settlement led to assimilation and eventual disappearance of the French language and cultural identity over time.
In Quebec, the population was largely homogenous and could more easily protect their linguistic identity from external influences. The early accommodation of the French-Canadians' language and identity after the Conquest, later codified into law, ensured the survival of the French language in Quebec and preserved Canada's unique dual identities.