Spanish, French and Italian are all Romance languages, the Roman part of the word Romance meaning, indeed, these languages are based on Latin - not because they are romantic! Hence Latin countries speak languages derived from Latin.

Not only Spanish colonies make up Latin America! Brazil, settled by Portugal and once center of the Portuguese Empire, is absolutely part of Latin America. French Guiana and the former British possession of Belize are often considered Latin American as well.

Chasteen in Born and Blood and Fire has attributed it to a public relations effort by Napoleon III in his campaign to legitimize the French takeover of Mexico in 1864. It fits in squarely with the nationalism-construction projects that Patrick Geary describes in the Myth of Nations, in that 19th century Europe is a fertile time for rational-sounding frameworks that attach the weight of all sorts of mythological lineages to then-modern countries and peoples. It's hard to predict which of these kinds of things will have currency for long; it's particularly interesting how Latin America endures as a concept despite not making much sense and being mostly de-linked from France in the public eye.

It's a pretty well-regarded textbook (I favor it because it's brief and it reads well, mainly), and I've not heard of other explanations.