They are armbands. The black mourning band is a "black arm band" or a "mourning arm band". They were the sign of an engineer; my father always wore them, necessary or not, as a symbol that, even though he was the boss, he was still ready to clear his arms to get oily and dirty as an engineer. It was like Bernard Weatherill, a Savile Row tailor, and Speaker of the House of Commons, always carrying a needle in his lapel.Never forget where you came from and what you are.
I wore stiff collars with studs all my working life. We had to, but we had to have specialist cleaners for the collars because they required extra starch and a gloss finish. We have wing collars for in court and cutaway collars for outside. Fitting a new collar, fresh from the laundry, is quite a skill, but after a few dozen goes you can do it with your eyes shut.