Is the war pilot obligated to fly the same plane daily or must he change the plane according to every required mission?
Of which type of pilot are you asking:


Student
Military
Private (hobbyist)
Corporate
Airline


I will be using the term licensed to loosely mean certificated, current, and proficient with the proper endorsements and documentations.

The short answer is no. They are not obliged/obligated to fly the same aircraft. The pilot can choose to fly in any aircraft they want. They will more than likely choose an aircraft that fits their mission.

The long answer depends on the mission. This might determine if the pilot gets the opportunity to fly at all. Or, what “mission” (job, employer, flight school, opportunity) they choose to seek. But, all aircraft fly a differently from each other. In the same make and model, the differences may be slight. In the same category and class, the differences may be very challenging. In different categories, classes, and types, the differences may be too great to overcome without specific training.

Students:
Most instructors will want their students to stick to one type of aircraft for their initial training (first 35-80 hours). That does not mean it will be the exact same aircraft. Large schools may have several identical aircraft of the same make, model, and avionics (when possible). The students will be expected to fly whichever aircraft is available at the time of the lesson. Although, the instructor can endorse the student to fly solo in any aircraft the instructor feels the student can handle on their own. This is essential in developing habits, muscle memory, and structure. 

After initial training, the student (then private pilot) is then only limited to flying the same category and class of aircraft in which he/she was trained. Although he student can seek training in another category and/or class. I flew both single engine land airplanes, and helicopters.

Private (hobbyist):
A private pilot can fly any aircraft for which they are properly licensed. The license will only stipulate category and class of aircraft for which a type rating is not required, type rating, and other flight category ratings. Type ratings are usually required for large and/or turbine aircraft. Some other ratings are based on flight categories (VFR vs IFR). There are some restrictions based on medical requirements. And, some flight categories will require specific endorsements (tailwheel, high altitude, high performance, complex, aerobatic, etc.) They can also fly in aircraft for which they are not properly licensed if a properly licensed Pilot-in-Command is also flying at the controls.

There is a class of hobbyist pilots who are very limited in the type of aircraft that they fly. These pilots need endorsements for any aircraft outside of their specific category, class, or type of aircraft.

As a practical matter, a hobbyist pilot may fly several aircraft based on availability. There are flying clubs and aircraft rental companies that give pilots access to aircraft as they wish. Provided the pilot has the proper license and is able to perform adequately on a flight evaluation. I regularly fly several different Piper Cherokee varieties, Cessna 150/2, 172 and 182 varieties, SportCruisers, Robinson R-22s, etc. Availability and cost both play a factor. 

Corporate &amp; Airline:
Generally, these types of pilots will hold type ratings in specific makes and models of aircraft. Since it is a pretty large expense to maintain currency of these type ratings, which ones they will be able to exercise might be limited. Which specific aircraft they fly within their license requirements will be up to the needs of their employer at that time. They may fly different aircraft day by day, or even flight by flight on the same day.

These pilots can still exercise the privileges of their private pilot certificate and many do. In which case, they are free to fly any aircraft for which they are licensed. Their employer may limit them to only fly on a for-hire basis for the employer, solely. That does not limit them from flying as a hobbyist. 

Military:
The military is going to determine which aircraft the pilot flies. It is the military after all. What the pilot flies is going to be based on the needs of the military. The pilot can make requests. But, being in the military means that the pilot has volunteered (in the U.S.) to do what they tell him/her to do. 

Generally, that is going to be mission and budget based. The military will train the pilot for a specific type of aircraft, for a specific type of mission. If the pilot or aircraft does not suit the mission, they use a different pilot. If the military chooses to use the original pilot, they send them back to training for that specific aircraft, for that specific mission. 

Both the trained pilot and the aircraft they are flying are very valuable assets. They are expensive tools that the military will handle with care (sort of). Mistakes cost lives and money.

What this means from a practical sense is that a pilot will not flip-flop between types of aircraft. There would have to be a very compelling reason to do so (like change of duty station or aircraft inventory). For the most part, the pilot will be assigned a specific individual aircraft unless the mission dictates otherwise. After all, it is their weapon. They must be proficient with it.

Think of it this way. You are not obliged/obligated to drive a certain car. You drive the car that fits your needs and budget. But then again, you are not obliged/obligated to work for an employer who may require you to drive a certain company vehicle.