One of the rituals from Victorian era’s cult of mourning: if a close loved one died, you’d order yourself some black bordered calling cards, note paper and envelopes. Your border’s width would be determined by how close you were to the deceased, how long you’d been in mourning, and how ostentatious you wanted to be.

Even newspapers followed this etiquette:

Vancouver Daily World, British Columbia, January 22, 1901:

The Guthrie Daily Leader, Oklahoma, September 14, 1901:

Anyway, to answer your question, I doubt it. Unless the guy’s brother was a big stickler for etiquette, pretty full of himself, or had a dark sense of humor, I don’t think you’d mail your own suicide note in a mourning envelope. And in that case it wouldn’t be a very funny comic.

(As an aside, in some places you couldn’t actually mail letters with full mourning borders. They were said to be easier to tamper with; edges could be cut and then coloured with dark ink, so ones that were tinted rather than full-black were also sold).

I’m pretty sure the joke is just that the guy isn’t very smart, and thinks the black border + the brother’s handwriting means his brother must be dead. Really his brother is just in mourning for someone else.