One may think of the French Netherlands as a holdover toponym, much like Austrian Netherlands or Dutch Netherlands (Spanish Netherlands) and thus archaic by that point. When the person was born, it may have been de rigeur for parts of the southern ten provinces of the Low Countries. The regions of Hainaut (or Hainault), Flanders and Luxembourg (or Luxemburg) bore the identification in encyclopedias from the early 19th century and had strong cultural and political ties to France even though parts of them were, at one point or another, under Austrian (or Holy Roman Empire) suzerainty prior to 1796 via treaties or inheritance--which could be kind of confusing as you can see in this 1778 map that has to indicate "French" vs "Austrian" Flanders, Hainau[l]t, etc. Flanders and Hainaut [edit: and Namur, I suppose, would be included in a general discussion] are of course in part in Belgium today, although not necessarily French-speaking across the entire area.

To make matters more difficult to parse, some of these areas became part of France and stayed that way. See this map to see how Hainaut, in particular, was partitioned and could put the region where your person was born into France proper despite being part of the "French Netherlands" at one time. The lines of authority and affinity are confusing and changed, but it either refers specifically to the French-speaking part of Belgium (Wallonia) and Luxembourg or the entire region of the Low Countries that isn't Dutch. I suppose context would tell you. 1891 is a little late to use such a term but not impossible, especially if it's referring to someone's birth during the Napoleonic Wars when France did control the former Austrian territories. Yes, it's kind of a mess. If it's an area in France now, it may have cleft across the border with Belgium or been part of the same area, especially with western Flanders or Hainaut. [edit 2: I sometimes see Cambesis also as part of historic "Netherlands" and that is definitely in France.]