"Facial tissue has been used for centuries in Japan, in the form of <a href="https://en.wikipedia.org/wiki/Washi" title="Washi">washi</a> (和紙) or <a href="https://en.wikipedia.org/wiki/Japanese_tissue" title="Japanese tissue">Japanese tissue</a>, as described in this 17th-century European account of the voyage of <a href="https://en.wikipedia.org/wiki/Hasekura_Tsunenaga" title="Hasekura Tsunenaga">Hasekura Tsunenaga</a>:"They blow their noses in soft silky papers the size of a hand, which they never use twice, so that they throw them on the ground after usage, and they were delighted to see our people around them precipitate themselves to pick them up.""
Do you mean Japanese people do or don't blow their nose with handkerchiefs? My first impression from your words is that they do. But upon rereading it seems like you mean they actually don't.
I used to carry a handkerchief with me in middle school and high school. I lived with my grandparents at the time and my grandmother made me carry one. My grandparents grew up in Taiwan when it was ruled by Japan, so they spoke fluent Japanese and retained a lot of Japanese customs. We had a drawer full of handkerchiefs and I remember picking through them to find a pattern that I liked. I didn't blow my nose with it but wiped sweat from my face in the summer or dry my hands after I washed them. I think it's a nice habit to have though I don't do it anymore. I may pick it back up one day; I remember sometimes finding a particularly pretty handkerchief and feeling nice just having something so pretty on me. It's a girl thing.
Regarding sniffing, I heard it's impolite to blow your nose in public in Japan. Personally, I'd rather people blow their nose and get it over with than to keep sniffing all the time.