I believe assembly language is a common noun; it is not a brand name and it is really a generic term for a set of languages that varies based on the hardware they support. Wikipedia uses it exclusively as a common noun as far as I can tell.
I feel your discomfort at writing
He wrote it in assembly.
because as a common noun it can cause confusion between assembly language and some other kind of assembly, notably the kind one attends in school. If it bothers you, you can include the "language" specifier to make clear the distinction: "He wrote it in assembly language." But it will probably be clear from the context anyway.
It took John two months to get his program working. He wrote it in assembly.
Most people who know anything about programming will understand perfectly well what kind of assembly is under discussion. If you intend the statement for a more general audience, which wouldn't know the difference, you can specify "assembly language" but probably that will still be insufficient to give those listeners an "a-ha" moment.