These days when one says "anniversary" one thinks wedding anniversary. Medieval people celebrated anniversaries too, but not wedding anniversaries and not birthdays. Rather, they celebrated death days.
Exactly how old a person was did not matter an enormous amount. Now we pass milestones (like driver's license or drinking) on particular birthdays, but medieval people rarely knew exactly how old they were. (This is not just medieval--I've been looking at census records, and before World War II a startlingly large number of adults had ages ending either in 0 or 5.)
Much more significant was the day of the year on which someone died. This is because anyone could be born, but a death was special. Accounts of saints might, very occasionally, mention something marvelous like the infant speaking even before birth, but they almost invariably gave a long, sad account of the saint's final days, the wisdom imparted to their followers, and their contrition for their sins. At monasteries, there was a special board that would be pounded if someone was dying, so all the monks could rush to the deathbed and join in someone's final moments.
The calendar of the year was marked by saints' days. Almost always the day that was special to a saint was the day on which that saint had died. For example, November 11 is not just Veterans' Day but Saint Martin's Day, because the saint was supposed to have died on that date.
One did not have to be a saint to have an anniversary day. Many nobles arranged that, after they died, monks would say special Masses in their memory, in perpetuity, on the day of their death.
I'm not sure how the flower forget-me-not was given its name. But it blooms in the spring and is a good reminder to recall those we loved who are no longer with us.