So I should point out some key concepts to understand military rank in the United States.

First, we must understand the the military of the US has been organized in different components even within each branch. The modern US Army, for instance, is made up of the Regular Army (standing military, aka active duty professional troops), Army Reserve, and the National Guard.

It was under the National Defense Act of 1920 that the Army was reorganized so that the the Army would be composed of the Regular Army, National Guard, and Organized Reserve component. The Regular Army had a distinct role in peacetime:

Each of the three Army components was to be so regulated in peacetime that it could contribute its appropriate share of troops in a war emergency.

The act acknowledged and authorized the historical practice of the United States: a standing peacetime Army too small to be expanded to meet the needs of a large war and reliance on a new force of citizen-soldiers when large-scale mobilizations were necessary. In contrast to earlier practice, training the National Guard and Organized Reserves became a major peacetime task of the Regular Army. To fulfill that mission Congress authorized a maximum Regular Army officer strength of 17,726 officers, more than three times the prewar number.

Thus, if you look at your link, Eisenhower held the rank of Major from August 26, 1924 until July 1st, 1936 when he was promoted to Lieutenant Colonel. His next rank is Colonel on March 11, 1941... in the Army of the United States.

What is the "Army of the United States?" Well, it is a "fourth" component of the Army, activated in times of war or crises (during the WWI period, it was called National Army) when manpower was needed (i.e. conscription). As a side note, the Army of the United States ceased when the draft was ended in the Vietnam-era.

Thus the first officers in the Army of the United States came from the Regular Army. They would retain their 'permanent ranks' in the Regular Army while holding 'temporary ranks' in the Army of the United States. All enlistments/conscription during war would enter the Army of the United States

This system was designed so that peacetime forces would administer/oversee the rapid expansion of a wartime military. After all, if you go from 400,000 troops to 10 million, you need experienced administrators, trainers, and leadership to make the massive increase in organization, growth in facilities/infrastructure, etc. even work. It may take years or even decades to master tactics and military leadership can't be easily obtained off the street, hence these experienced individuals were needed in the massively expanded military.

Thus, Eisenhower was made a Colonel in the Army of the United States, and from there rapidly rose in rank. This was again, both a sign of his leadership abilities, and also an acknowledgement that you needed more leadership as the military expanded rapidly. The growth of the US military to a peak of ten million with nearly 100 divisions and thousands of warships meant you needed more generals and admirals than during peace.

Hence why you see his Regular Army rank goes up during the war to a 2-star general, while he was a 4-star general in the Army of the United States. When the war ended and the Army of the United States was disbanded, his rank reverted to the 4-star General rank of the Regular Army. His 5-star rank was made permanent by Act of Congress in Public Law 79-333 here, and thus he assumed that rank after the war in the Regular Army.

One other aspect: in case you didn't catch it, but Congress actually directs how many officers are available to be generals/admirals at any time.

US Title 10 governs all aspects of the military, to even include who can be in command of an aircraft carrier.

The laws in fact govern the max number of generals/admirals:

The number of general officers on active duty in the Army, Air Force, and Marine Corps, and the number of flag officers on active duty in the Navy, may not exceed the number specified for the armed force concerned as follows:

(1) For the Army, 231.

(2) For the Navy, 162.

(3) For the Air Force, 198.

(4) For the Marine Corps, 62.

Title 10 even governs how many 4 star generals/admirals:

(1) in the Army, if that appointment would result in more than—

(A) 7 officers in the grade of general;

(B) 46 officers in a grade above the grade of major general; or

(C) 90 officers in the grade of major general;

(2) in the Air Force, if that appointment would result in more than—

(A) 9 officers in the grade of general;

(B) 44 officers in a grade above the grade of major general; or

(C) 73 officers in the grade of major general;

(3) in the Navy, if that appointment would result in more than—

(A) 6 officers in the grade of admiral;

(B) 33 officers in a grade above the grade of rear admiral; or

(C) 50 officers in the grade of rear admiral;

(4) in the Marine Corps, if that appointment would result in more than—

(A) 2 officers in the grade of general;

(B) 17 officers in a grade above the grade of major general; or

(C) 22 officers in the grade of major general.

There are notable exceptions: joint duty, like being commander of CENTCOM, will be a four star and thus branches can exceed caps when they have individuals from their ranks promoted to joint jobs. For instance, currently the US Navy has 9 four-star Admirals. The "statutory" 6 are:

• Chief of Naval Operations - ADM Richardson
• Vice Chief of Naval Operations - ADM Moran
• Director of Naval Nuclear Propulsion - ADM Caldwell
• Commander, U.S. Fleet Forces Command - ADM Davidson
• Commander, U.S. Pacific Fleet - ADM Swift
• Commander, U.S. Naval Forces Europe - ADM Foggo

The other 3 are in joint jobs:

• Commander, U.S. Pacific Command - ADM Harris
• Commander, U.S. Southern Command - ADM Tidd
• Commander, US Cyber Command/Director, National Security Agency - ADM Rogers

Thus, at the general/admiral level, you are actually promoted into your job. For instance, the Chief of Staff of the Army is always a 4-star general. One can skip having a 3-star job and go straight to that position and assume the rank of 4-star. Once you leave that job, your rank reverts to your previous rank... unless you go into another 4-star job or retire.

So in the case of Eisenhower, he assumed rank quickly because he took on the roles and responsibilities of what a 4-star was to be in charge of as the Army of the United States expanded rapidly. His permanent rank, in the Regular Army, rose more slowly. And his 5-star rank was made permanent after the war by an Act of Congress.

edit: typos and clarification

/u/technique_only gave a great, comprehensive answer, but I wanted to add one thing about Eisenhower and his meteoric rise with respect to why he merited promotion to such a senior rank and billet in the first place:

Eisenhower can, in some respects, be seen as an example of failing upwards. This is not to say that Eisenhower was a failure as a commander, by any means... but you have to remember that he spent most of his peacetime career as a staff officer, and had no combat experience and only a few months of real command experience when the U.S. entered WWII. A big part of Eisenhower's career success is owed to the fact that he had ingratiated himself to the right people during his staff assignments: Douglas MacArthur, George Marshall, and even John Pershing all considered Eisenhower a friend and a trusted junior officer. In addition to the exceptional competence, alacrity, and work ethic he exhibited with regards to his peacetime duties; Army officers, much like voters in 1952, did very much "like Ike".

Eisenhower's inexperience and relative ineptitude as a tactician and battlefield commander, on the other hand, directly resulted in a massive number of Allied deaths in what should have been slam dunk victories against a beleaguered German army in North Africa during Operation Torch, and likewise during invasions in Sicily and Italy... basically, everything before Operation Overlord was a fucking shitshow.

So then why did Eisenhower keep getting promoted over better tacticians like George Patton (a personal friend who had been senior to Eisenhower for decades) and Bernard Montgomery (who was already a battle-hardened Marshall by the time the U.S. arrived in Europe)?

The short answer is that he had a gift for the unique and unprecedented political demands of a coalition commander of that scale, whose duties involved smoothing relations between not only competing Generals from other nations, but also headstrong world leaders embroiled in a perpetual dick measuring contest. No other military officer on the planet could have mollified FDR and Churchill with regards to the looming question of Charles de Gaulle's role in both the Allied war effort and post-war Europe, for example. While his lack of command experience left him ill-prepared for waging war, his unique experiences weaving in and out of the War Department and the White House, at the right hand of legendary commanders and world leaders alike, left him uniquely well equipped to hold together the most massive coalition army in human history.

Thus, expediting Eisenhower through the ranks despite his failures was dual purpose: The rarified position of Supreme Allied Commander not only allowed Eisenhower to put his political acumen to good use, but it allowed more skilled battlefield commanders, like Patton, J. Lawton Collins, and Matthew Ridgeway, to direct troops on the ground.

Eisenhower's rise was unusually fast, even by WWII standards. Jean Edward Smith details his career trajectory really nicely in "Eisenhower: In War and Peace".