Is water the only substance freezing at 0°C? When reviewing with my son his science homework (12 years old = ~7th grade, 5ème in France), he told me that "when the phase transition graph shows a plateau between phases then the substance is pure. For instance, if the plateau is at 0°C then we know this is pure water".
I told him that this is not correct, at last the second part, because we do not know that only water freezes at 0°C. He then showed me his copybook where there was a statement from a book (translated):

The temperature plateau enables us to identify the chemical substance. (...)
  If we observe a temperature plateau at O°C during a fusion or
  solidification, or at 100°C during a vaporization, then the chemical
  substance is pure water.

Is this correct? Are there physical aspects which guarantee that no other substance freezes at 0°C? (or more generally - that two substances do not have the same temperature of solidification?)
Note following a comment: for the sake of the question, we can assume a sigle pressure at which all substances are measured (specifically 1 atm for the question as worded).
 A: Of course not. Depending of pressure applied on the substance you can force almost any substance to freeze at 0c. 
A: Electrical forces between molecules determine the temperatures at which a substance freezes and boils; the stronger the forces, the higher the temperature. The freezing point of a substance is the temperature at which the force of attraction between its molecules becomes strong enough to overcome the energy of motion that its molecules have when the substance is in its liquid state. 
Now what makes water special is that it is a polar molecule means partially negatively charged at one end and positive at another and this polarity helps a single water molecule to form hydrogen bonds with other two water molecules. And hydrogen bonds are stronger than most of the bonds found in other chemicals. That means comparatively more energy is required to break this bond. Impurities and pressure variation changes the strength of the bonds, further affecting the freezing and the boiling point. So basically its the strength of the bonds between the molecules of a chemical that decides its boiling and freezing point.
Now for other chemicals the strength of the bonds is increased by 
1. Increase in the  number of carbon atoms, 
2. Increase in molecular weight, 
3. Increase in symmetry meaning decreased branching 
4. and increase in relative strength of the four intermolecular forces ie: Ionic > Hydrogen bonding > dipole dipole > Van der Waals dispersion forces. The influence of each of these attractive forces will depend on the functional groups present.
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Impurities fills the voids between molecules and hence make it denser... and since they possess charge so they are bonded with weak force, therefore to break these additional bonds additional energy is required hence boiling point increases... boiling point is a temperature threshold point where the surface particals attains that much upward pressure that they can overcome the atmospheric pressure and hence they leave the surface. Pure water has fixed boiling and freezing point and if this point changes during transition phase then the water surely has impurities in it. 
For example sea water has increased boiling point at atmospheric pressure ie 100.7 degrees C due to added impurities. 
........
Space is extremely low pressure and extremely low temperature zone. When the astronauts take a leak while on a mission and expel the result into space, it boils violently. The vapor then passes immediately into the solid state (a process known as desublimation), and you end up with a cloud of very fine crystals of frozen urine.
But when that water boils, remember how much more volume gas takes up than liquid, and how much farther apart the molecules get. This means that immediately after the water boils, this water vapor — now at effectively zero pressure — can cool very rapidly! 
The quick reduction in pressure (going from having water on top of it to just air) will cause a rapid boil, and then the quick action of the extremely cold air on the water vapor will cause the formation of frozen crystals: snow!
