I'd guess it's because it's so much quicker to say "assembler" than "assembly language." I remember a lot of people saying "ML" in the 80s. I liked "ML." Nice and short, and it sounds like it could be one of Superman's relatives.
"Assembly language" is long and awkward. It sounds like a term that might have come out of the UN. "Assembler" has a nice "blood and guts" feel that matches the experience of low-level programming.
The usage of "assembler" to mean "assembly language" has been around for decades. "Written in assembly language" just barely beats "written in assembler" in a Google fight, so on the usage front I'd say either is valid. "Code Complete 2" uses the term "assembler" in the description of languages section.
You get many historically interesting pages if you search for "written in 68000 assembler," "written in 6502 assembler," etc.
The usage is mentioned on wikipedia.
Note that, in normal professional
usage, the term assembler is often
used ambiguously: It is frequently
used to refer to an assembly language
itself, rather than to the assembler
utility. Thus: "CP/CMS was written in
S/360 assembler" as opposed to "ASM-H
was a widely-used S/370 assembler."
Words often have multiple meanings. English is not assembler.