It shouldn't. 'Assembly' is the textual code, 'assembler' is the tool used to convert the text to machine code. I don't think the name is worth an argument, but your suggestion is no more correct than that is already in the title.
Considering I spent my youth writing assembler code, I find it pretty upsetting someone from the internet lecturing me on what I was deeply involved with is called. I think I know what I was writing.
I don't think anybody is claiming it's not called assembler, just that it's also / mostly called assembly. fwiw I wrote some [machine code via mnemonics] in the '80s (admittedly not lots) and assembly sounds more normal to me.
It's a dangerous business, going onto the internet. You step into a site, and if you don't keep your feet, there is no knowing what you might hear.