\mathscr and similiar ones don’t refer to a specific font. If you look at different math fonts like Cambria Math, Asana Math, XITS, Minion Math etc. you see that they are designed with different ideas in mind, hence they look different.
\mathscr just request symbols that look script-like. Of course every font designer has had a different idea what “script-like” should be for his font, thus you get different results. This means that there is no precise definition what
\mathscr should look like. Some fonts may not have such symbols at all, some may have more than one, like XITS where you can get a “calligraphic” and a “script” like looking variant of the font.
The actual symbols you get depend on what math font is loaded via
\setmathfont in case of
unicode-math and LuaTeX or the more traditional NFSS in case of PDFTeX. With NFSS setting the math fonts is more involved. So usually this is done by loading an appropriate package via
\usepackage that switches to the fonts in question.