Yes, if a dog is allergic to chicken, then it probably means to all parts of the chicken - including the fat. A sometimes-exception is raw chicken - that is, a dog that is allergic to chicken kibble is not always allergic to raw chicken. That's not as crazy as it sounds, since the cooking process alters proteins - so it is possible for some dogs to be unable to tolerate cooked chicken, but to do fine with raw. Doesn't happen always, of course, but does sometimes. I would doubt this would apply in the case of chicken fat added to non-chicken kibble though, as it is still going to be cooked. I guess there is only one way to be certain for any individual dog though, and that is to try it.
As to the second part of your question... yes, an increasing number of dogs are allergic to chicken. This is because it is overfed to them (no surprise - it is the cheapest and most readily available meat source around). Allergies will usually only develop when a dog (or person) is constantly exposed to the same thing. Weell - not only is chicken in a large number of dog foods, but most people feed the same food to their dogs every meal, every day, for extended periods of time

Even for those who rotate foods regularly, that can mean weeks on the same thing (and for some people, it means months or even years). That is the surest way to induce an allergy. The same thing used to be the case for beef, when that was the most common meat in dog food. And the same thing WILL happen if/when lamb or turkey or fish or any other meat becomes overfed (either en mass, or individually).
As for why dog food manufacturers throw chicken fat into so many foods - the answer is the same : because it is cheap and readily available. You have only to look at the low cost of kibble to understand the economics (it costs less per lb than most of the 'better' ingredients used to make it... despite processing, manufacturing, marketing, plant, machinery, staff and shipping costs. How can you realistically expect that most of what is in there isn't junk?).
Having said that, there are a number of options out there that do not contain chicken anything. As a very broad generalisation, foods that use oils rather than animal fats (chicken or otherwise) are often much better anyway - you could as easily choose one of those.