What happens to bread when it is done
Yes, there is something particular what happens at a temperature in the mid-90s. Not all details of it are proven, but the major outline is, and the hypotheses about the details are solid enough to make it into textbooks.
Starch is contained in tiny granules, a few micrometers in diameter. When heated in the presence of water, there is a specific temperature at which these granules burst. The molecules of starch come in contact with water and the water molecules get lodged in the nooks and crannies of the much larger starch molecules. This process is called gelating.
You can observe it easily on the macro level. Just cook a bechamel or starch pudding on stovetop, stirring constantly. The liquid will stay rather thin until all will thicken at once, just before you see the first bubbles of boiling. This is when the starch gelates.
The same thing happens in bread too. This is why you want to heat the bread to this temperature. If you don't, you will have raw starch inside, which doesn't taste well.
The exact temperature at which this happens varies a bit with the type of starch. It is not the same for rice and wheat, for example, and I think that it is also a bit different between different wheat cultivars. But the range within this variation occurs is not so wide, all references I have seen move somewhere between 94 and 98 degrees Celsius. So the recipe author just picks a temperature he knows to work for the flour used in the bread, maybe also accounting for some additional heat transfer after taking out of the oven.
Can you use temperature as an indicator for doneness
The theory says yes. My personal experience also says yes. Why did you feel that your bread was too doughy? There are different reasons why this could have happened. You could have measured it wrong (with the probe being too close to the surface, where the temperature is higher). You could have cut it too early. (Bread is always doughy before the first starch retrogradation, which occurs maybe 1 hour after baking). It is also possible that the bread was actually done in the sense of gelled starch, but that the recipe produced a rather moist bread and that you have grown accustomed to dry breads if you normally bake your breads for a very long time, so your brain perceived the unaccustomed texture as "not right". Or it is possible that something went wrong with the leavening, making the bread too dense. Dense bread is always doughy, you cannot bake the moisture out of it.
technical criteria for bread doneness
There are two big chemical changes which happen to bread while baking. The proteins in bread (the gluten) have to harden. Before that, they are soft and pliable. At some temperature, they become rubber-like. The hardened gluten gives the bread structure.
The second change is the starch gelation I explained earlier. When this happens, the liquid part of the dough (dough consists of a liquid phase suspenede in the elastic gluten mesh) thickens. Gelated starch gives bread a fluffy, soft body.
As the starch gelates at much hicher temperatures than proteins denature, bread is taken out of the oven when the starch is done.
The third step is the starch retrogradation. In retrogradation, starch loses the water which it took during gelation. There are three big stages of it, after each the texture changes drastically. The first happens at about an hour after getting out of the oven. This is when the bread is considered done by textbooks. In practice, there are many people (including myself) who like the taste of the moist hot bread just out of the oven, and they consider it done at the previous step. The second happens after about 24 hours; after it, the bread is considered stale. The third step takes several days, and after it, bread is considered inedible, because it becomes hard as wood.
So technically, bread is considered done after it has been baked to gelation temperature and then left alone for 1 hour.