How did "Flemish Cape" get its name?

Maybe you are asking for the Flemish Cap, which is not a "cape"?

https://en.wikipedia.org/wiki/Flemish_Cap

Does it have anything to do with Dutch piracy in Newfoundland?

https://www.cambridge.org/core/books/abs/war-and-trade-in-eighteenthcentury-newfoundland/problem-of-piracy-in-the-newfoundland-fishery-in-the-aftermath-of-the-war-of-the-spanish-succession/F7EAAF94F91F6CF3F95F74699DC5A771

Or possibly it has something to do with the Dutch seizure of St.Johns during the Anglo-Dutch Wars? As you can see, I'm only guessing.

https://en.wikipedia.org/wiki/St._John%27s%2C_Newfoundland_and_Labrador#History

I found Flemish Cape as well as Flemish Cap. See this site of New York Times.
http://query.nytimes.com/gst/abstract.html?res=9B0DE3D8153AE533A25757C2A96F9C94609ED7CF
BRAVE FISHERMEN ALARMED.; THE SEA SERPENT AND A HUGE SHARK OFF FLEMISH... - Article Preview - The New York Times
Confusion between Flemish and Dutch is not unusual in a medieval context. Ships from Flanders and ships from the Zealandese islands won't have been perceived as so very different. But after the Dutch revolution there must have been a more marked dividing line. Not very sure that the "Flemish cap" got its name from Dutch pirates.

Etymologically speaking, the term "Flemish Cap" is a misnomer from the Flemish/Dutch term "De Vlaamse Kaap", which refers to Flemish Kaapvaart (or trawling in and around the grand banks and international waters nearest Canada in the 1600s).

http://untreaty.un.org/unts/60001_120000/23/13/00044634.pdf

English speakers bastardized the word "Kaap" into "Cap", given the similarity in pronunciation. However, "Kaap" in fact translates to "Cape"... just as "Kaapstad, Zuid-Afrika" translates to "Capetown, South Africa".

http://nl.wikipedia.org/wiki/Kaapstad