Most people know that we read Az Yashir on Shabbos Shira and we put out bread for the birds. However this is the only Shabbos that has a special name simply because of what we read in the Parsha. (Sefer Hatodaah)
Shabbos HaGodol, Shabbos Shuva, and others are not named after what we read in the Parsha.
ספר התודעה asks a similar question, and gives a beautiful answer:
Not only שירת הים is in this פרשה; many other ideas are.... Nonetheless, the Jewish people has designated that this שבת be named specifically after the שירה, for this שירה was for Jews throughout the generations as if they were then saying it. Why so? Because whereas ה׳ spoke the entire Torah, and the Jews listened, here the Jews spoke and ה׳ and all His hosts listened....
For with the strength of the שירה that they then sang, they planted שירה and joy in the heart of every Jew, for all time; every time they are saved from their enemies and granted reprieve from their troubles, their hearts sing... to ה׳ Who saved them.
On another note, ספר אהל משה, by Rabbi Moshe Rosen of New York, notes that this week should not, by rights, be called שבת שירה merely because we read the שירה. After all, we read the שירה on a daily basis, so what makes this day special? Further, it can't be called שבת שירה because we commemorate the miracle of the sea now — we do that on the seventh of פסח, the anniversary of the crossing.
He answers that the reason the Jews sang at that point — even though a trip through the deadly desert was still before them — was that they trusted that ה׳ would take care of them. "ויאמינו בה׳ ובמשה עבדו, אז ישיר...". So we refer only to this day as שבת שירה, as this is the only day on which we also read all the troubles that they went through in the desert. (Every day, when we read the שירה, we read it alone, without recalling the hardships of the desert.) When we read it this week, it strengthens our own בטחון. (We do not call the seventh of פסח by the name יום שירה because we do not need to strengthen our בטחון, as we are שרויִם [בשמחה] and have בטחון.)
However this is the only Shabbos that has a special name simply because of what we read in the Parsha.
This is also the only parasha that has an extremely visible, long, brick-formatted, column-widening song right in the middle of it. I think the specialness of the name matches the specialness of the Torah reading. My only source for this is my intuition.