Why does the mixture's temperature change while phase transition?
The concentration of water decreases as the mixture cools, so one is left with fairly pure ice, freezing at ~273 K, and pure(r) ethanol, freezing ~159 K (well, the eutectic mixture, about 93% ethanol). Since the ratio of the components are changing as it freezes, there is no distinct freezing point (unless you start with the eutectic mixture). See the phase diagram for ethanol-water mixture. Instead, the slush mix has a range of solidus and liquidus temperature. BTW, this is a method to make apple jack or brandy by freezing out water from cider or wine.

The same is noticed with solder, where it crystallizes gradually and dully, except for the sudden freezing of a eutectic mix (~63% Sn, ~37% Pb), making a shiny joint.