In Japan, everyone gives carnations on Mother’s Day. If you ask why, they shrug. Some might say that carnations are a symbol of mothers. Ask why, and they'll shrug again.
The thing with "traditions" in Japan is that a lot of them are imported. And, more often than not, the country from which the custom was adopted has itself long stopped doing it. The traditional school uniform for boys here is a high collared black jacket with buttons down the front. This came from Prussia over a hundred years ago. The sailor uniforms, too, were adopted from, I think, the clothes a British prince was wearing a century ago and the principal of a school I worked for adopted it as the school's uniform. It took off from there, becoming standard in Japan.
Mothers Day was also imported, most likely from the US, introduced in the postwar years by Christian missionaries. There was a different Respect Mothers Day that was the birthday of the Empress before the war, but like many of the pre-1945 holidays that were related to the Emperor they got thrown out or repackaged.
So, again, I suspect that Christian missionaries brought the custom of giving carnations to mothers on Mother’s Day. Why carnations? Because legend has it that carnations started to grow where the Virgin Mary's tears fell when she saw Jesus carrying the cross pass by.
I had never heard of this and looked up the Stations of the Cross to see if there was any mention of it. The 4th Station of the Cross (Traditional) is where Jesus meets his mother. Nothing about carnations or flowers there, so I still don't know how that all got started.
From Wiki: "Out of the fourteen traditional Stations of the Cross, only eight have a clear scriptural foundation. Stations 3, 4, 6, 7, and 9 are not specifically attested to in the gospels (in particular, no evidence exists of station 6 ever being known before medieval times) and Station 13 (representing Jesus's body being taken down off the cross and laid in the arms of his mother Mary) seems to embellish the gospels' record, which states that Joseph of Arimathea took Jesus down from the cross and buried him."