There are only a few areas of law of which I am aware that U.S. law treats people who are engaged to be married differently (although perhaps with more thought I could expand the list).
Fiance(e)s come under a special immigration status when applying for a visa.
There is a body of law related to whether an engagement ring is an absolute gift or is conditional upon being married (this is not uniform from state to state and I don't recall what the majority rule in those cases is under the common law). In South Carolina, ownership disputes over engagement rings are litigated as breach of promise to marry actions.
Pre-marital agreements governing a future marriage can be made by people who are engaged (although a post-nuptial agreement is also equal in effect in most cases).
People who are engaged, like spouses, are generally considered to be in a "confidential relationship" with each other which imposes higher duties with respect to fairness in their dealings with each other than strangers, but generally less high duties than fiduciaries.
While not strictly arising from the status of being engaged, adult cohabitants are generally agents for service of process of each other at their shared home, and are often considered to have a legally significant relationship for purposes of domestic violence statutes (usually related to either domestic violence crimes or temporary restraining orders).
I am not familiar enough with the law of France to fully answer the balance of the question, which someone more familiar with that law can expand upon.
But, there is similarly a special immigration status in French law for a fiance(e).
And, French law, in a flourish so romantic it could scarcely be any other country's law, also allows people who are engaged a right to marry after one of them has died in certain circumstances:
"The legislation that allows posthumous marriages stems back to when a dam burst in 1959 and killed 420 people in southern France." It was most recently invoked in 2014 when a grieving French woman was granted permission by the French President to marry her former fiancé, who tragically died in 2012, just a month before they were due to wed. To be eligible the bride to be had "to convince the President of France that her’s was a special case and that her love for Michael went beyond the grave. It took four letters to the president and 20 months of waiting, desperately hoping for a positive response." The President's discretion in this matter is somewhat similar to the pardon power in U.S. law.
This French law was also invoked in 2009. The law in question is set forth at Articles 171 of the French Civil Code. In English translation this states:
Article 171 The President of the Republic may, for serious reasons,
authorize the celebration of the marriage if one of the future spouses
is dead providing a sufficient gathering of facts establishes
unequivocally his consent. In this case, the effects of the marriage
date back to the day preceding that of the death of the spouse.
However, this marriage does not carry with it any right of intestate
succession to the benefit of the surviving spouse and no matrimonial
regime is considered to have existed between the spouses.
I am aware of one documented case where a court entered a post-humous marriage in the United States between people who were engaged, but I am not familiar with any legal authority actually authorizing that action.