Earlier quoted context omitted.
It's actually not all that surprising if you approach it from a different angle. Consider that light is nothing more than electromagnetic radiation; it is generated or absorbed only by changes in electromagnetic potential. Now also consider that electron orbitals in molecules are discrete and quantized: you can be in this orbital (with such-and-such potential energy) or in that orbital (with a slightly different pote…
How are they able to see the signature water gives off through all the noise that other types of molecules must be sending back? Isn't there even more noise to deal with when doing this over even bigger distances?
Stuff being in the way will of course diminish the signal (by direct absorption or by scattering it). But space is pretty empty. The regular air on Earth contains ~10^19 particles per cm³, and it can take several km of air to start seriously interfering with light. A dense portion of space contains something like 10^4 particles per cm³, and a less dense region maybe 10^-4 particles per cm³. So, once you leave the atmosphere, the noise that's destroying the signal is simply far, far, far weaker than the signal.
Now for your first question. As I mentioned in my first comment, the spectral lines arise from different orbitals. What this means is that there isn't a single line associated with a molecule, but actually several lines. So if you think you see a line from one molecule, you should be able to find the other associated lines, and the more of the lines you can find, the stronger the evidence.
As a couple of people have noted, this isn't a perfect process, so you can get into arguments if your identification is actually correct or not.