The mishna says that they would read Hallel by each of the three groups as they brought their korban pesach. Tosafos cites the Tosefta that it was the Levi'im who would read the Hallel, not everyone. Tosafos asks that if it was the Levi'im who would be reading the Hallel, it should have a status of שירה and would be a violation of the rule אין אומרים שירה אלא על היין. Tosafos answers that the halacha of אין אומרים שירה אלא על היין is a halacha in being "makriv" the korban. The shira which complimented the burning of the korban on the mizbeiach was done specifically at the time of the nisuch ha'yayin. Here the Hallel is not being said at the time of the hakrava, rather we are talking about Hallel at the time of shechita. On this type of shira there is no requirement of אין אומרים שירה אלא על היין.
The Brisker Rav (Hil. Chanuka) cites the Rambam who says that Hallel is a תקנה מדברי סופרים, whereas the Magid Mishna writes that it was a תקנת נביאים which is the language of the gemara as well. The Brisker Rav explains that there are 2 dinim in Hallel. There is a Hallel of שירה such as the Hallel said at the time of the eating of the korban pesach - כליל התקדש חג, as we see in the gemara 95b. There is also the Hallel which is a קריאה such as the Hallel done on every Yom Tov in shul. Hallel as a שירה is a תקנת נביאים but Hallel as a קריאה is מדברי סופרים. Regarding the Bracha on Hallel, we only make a bracha (מעיקר הדין) on the "full" Hallel because only the full Hallel has status of Hallel. But when Hallel is being said as a shira, there is no specific amount that must be said, therefore they could make a bracha on it even if it wasn't completed. The Brisker Rav explains that the Hallel at the time of shechting the korban pesach is a din in shira, therefore it wasn't necessary for the 3rd group to complete it as the Mishna explains. The Brisker Rav concludes by citing a Tosafos HaRosh that there are 3 times where we say Hallel without wine, which are exceptions to the rule of אין אומרים שירה אלא על היין. The 3 times are: 1. Hallel of shechitas HaPesach. 2. Hallel by eating the Pesach. 3. Hallel by war. Why don't they also count every Hallel of Yom Tov which are also exceptions? The Brisker Rav explains that only when the Hallel is being said as a shira do we regard it as an exception when it is done without wine. When the Hallel is being said as a קריאה, of course it is said without wine.
Based on this, Tosafos would also categorize Hallel of shechitas HaPesach as a שירה not a קריאה, yet since it is not done at the time of hakrava, we don't require it to be done on wine.