I think the piece of material could also be called a bib

What is a Plastron?
The Plastron (French “front shirt”) is a bib, breast trim or insert on the shirt. This insert can be found on historical uniforms and traditional costumes. Originally the Plastron is a breast-covering form of the tie which was worn by men around 1860 and is still known today as the Ascot-Tie in riding clothing.

Around 1900, a lingerie stand-up collar in women’s fashion was also called a plastron. This plastron later developed into a sewn-in or sewn-on insert on blouses and dresses, in fabric with decorations (such as tucks or frills). (from Reclams Mode & Kostüm Lexikon by Ingrid Loschek). Today, the Plastron shirt can be found in men’s and women’s clothing.