I recently watched Your Name again and translated the text seen on screen such as the chalk on the whiteboard or cell phone message exchanges.
In one of the first few scenes of the film, Mitsuha’s teacher (who is Yukari Yukino from Shinkai’s other film Garden of Words by the way) talks about tasogare-doki, or “twilight”, which originated from the word tasokare meaning “who are you?” If you look at the characters for tasokare, it can literally be read as “who is he?”; a contrast to the note that Mitsuha was looking at in the beginning of the scene which says “who are you?” as written by Taki.
The note on the chalkboard is actually poem #2240 in the book poetry anthology book titled Manyōshū. It reads:
『誰(た)そ彼と われをな問ひそ 九月(ながつき)の 露に濡れつつ 君待つわれを』
tasokare to ware wo natohiso nagatsuki no tsuyu ni nuretsutsu kimi matsu ware wo
Which translates to: “do not ask me who I am, for I am the one who waits for you, drenched in the dew of September.”
This poem says a lot about the Your Name story, particularly the relationship between the two main characters. In fact, September is the month the story begins. The dew is a symbolism for the speaker’s longing and sadness after waiting for their lover for so long. Now let’s return to the words that were mentioned earlier, tasogare-doki and tasokare.
The former refers to the twilight, where it’s neither sunny nor dark. When people encounter each other during this time of day, it’s too dark to make out each other’s face. Therefore, people would ask… “who are you?”. Hence the etymology.
This poem is actually played out in a later scene of the film, where Mitsuha finds Taki in Tokyo, but unfortunately, the latter doesn’t recognize her. And another time during the climax of the film where the two finally get to meet each other one last time within the twilight before the epilogue of Your Name.