First of all, let us define 'victory' in this scenario as the axis countries (Germany and Japan, mainly) successfully holding the majority of the territory they captured and creating a stable platform for further expansion, if desired.
My friend's argument boils down to the claim that the early successes of these countries was largely due to the fact they had been /preparing/ for war in one respect or another (by building armies, weapons, etc.) and caught the invaded nations by surprise.
He goes on to insist it was this surplus, more than any tactical superiority, that contributed to their initial notoriety and that once the allied war industry was in full swing their number was essentially up. According to him arguments of military prowess, cunning battle tactics, and superior technology on behalf of the axis powers are largely the effect of postwar culture making them seem bigger and menacing than they actually were in order to make the allied nations that much more impressive for defeating them, but that in actuality the rigid authoritarian hierarchy of the German and Japanese empires made them either creatively or bureaucratically stunted and ill-equipped to adapt to the allies once they got their feet on the ground.
So in essence, how much of this is true? Were the Axis countries ultimately bound to fail in keeping their territory, and has their threat (Not their atrocities mind you, but their 'threat' as a military enemy) been exaggerated in the decades since to make the Allies seem like the underdog?
Your friend is correct, Germany, Italy and Japan had inferior economic power and population vice-verse the allies. Which doomed them in a long war of attrition against the allied powers.
By Paul Kennedy's calculation in 1937 US alone had around 42% of the total war making capacities on the planet, the Soviet Union and Germany each had around 14%, while the Japanese had a mere 3.5%.
Once properly mobilized, the allied countries can field far larger, and more powerful armies, navies and airforces than can the Axis powers. A good example of this is that Nazi Germany never had the resources to establish a capable strategic air force because the cost to building large bomber airplanes were prohibitively high (the resources to make a single bomber like a B-17 is many times that of a tank). The Nazis therefore tried to bomb the UK during the Battle of Britain with what amounted to medium bombers.
The US and the UK by 1942 on the other hand were capable of putting over 1000 heavy bombers above the skies of Germany. This is just one the many examples of superior allied material advantage, other examples include the fact that the US army was used motorized vehicles for logistics whereas the Wehrmacht used horses because it had neither the industrial capacity nor the fuel to use trucks to transport supplies. The budget for the Manhattan project alone was equivalent of an entire year's spending for all military activities by the German Reich.
The advantage of the Axis power is their early rearmament, early advantages in military leadership and doctrine, incredible levels of incompetence on the part of their enemies (such as Stalin refusing to believe that the Germans were invading even after the Whermacht tanks crossed the border) and sheer luck which allowed them to temporarily achieve victory over their opponents. They extracted value from their early war military advantage, to a dramatic degree and fueled postwar literature and pop culture impression of a Germany military much more powerful than it actually was.
However, on the long run, Germany was a medium sized European country fighting against two sub-continental empires and the world's largest maritime colonial empire. As long as its opponents refuse to make peace, it's defeat was always inevitable because of its economic inferiority. The same is also true with Japan, only to a larger degree.
Source: Wages of Destruction by Adam Tooze
Rise and Fall of the Great Powers by Paul Kennedy
War of the World by Nail Ferguson