This question is mentioned by Rav Elimelech Biderman shlita in this week's Torah Wellsprings - Beshalach 5784 (pp.16-17).
He writes there:
After maariv on Shabbos Shirah, Rebbe Bunim of Pshischa zt'l repeated a question he heard from Reb Yitzchak Vorka zt'l. Reb Yitzchak Vorka asked, "Why is this Shabbos
called Shabbos Shirah? We generally don’t name the Shabbos after what we read in the
Torah. Last week's Shabbos (parashas Bo) isn't called 'Shabbos Yetzias Mitzrayim,' and next week's Shabbos (parashas Yisro) isn't called 'Shabbos Matan Torah.' Why, then, is this Shabbos called Shabbos Shirah?"
Rebbe Bunim answered that it is because the Shirah is written in the sefer Torah לבינה אריח גבי על. [Az Yashir is written in the sefer Torah in a song format, with large spaces after each phrase.]
Then Rebbe Bunim said with hislahavus, "Every Yid must be very happy on this Shabbos!"
The Ramasayim Tzofim zt'l (Tana d'bei Eliyahu Zuta 16) offers the following explanation:
It is based on the Shlah Hakadosh, who says that the ink of a sefer Torah represents the neshamos, the good deeds, the emunah, and the daas of the Jewish nation. The white of a sefer Torah represents the physical bodies of Bnei Yisrael. He explains that when we read about yetzias Mitzrayim in the Torah, the typical
Yid doesn't experience the joy of yetzias Mitzrayim, and when we read about matan
Torah, the average Yid doesn't experience the incredible joy of matan Torah. Tzaddikim do experience the joy; however, the average Yid doesn't. This is why we don't call last Shabbos "Shabbos Yetzias Mitzrayim" and next Shabbos "Shabbos Matan Torah."
However, when we read Az Yashir, every Yid can experience the joy of kriyas Yam Suf. There are many white spaces between the stanzas, which means that the bodies of
Bnei Yisrael also experience it. The human mind and heart experience the joy of kriyas Yam Suf. It can therefore be called Shabbos Shirah. It is a joy that everyone can experience.
This is why Rebbe Bunim of Pshischa urged every Yid to be happy on this Shabbos. This is a joy that everyone can relate to.