The history of the assault rifle boils down to countries trying to find an intermediary cartridge between what's considered a machine gun, rifle, battle rifle, etc that would be acceptable to use in an automatic design that is deployable as a rifle. The Federov Avtomat (Russian) is sometimes considered the first assault rifle, or at least an important predeccessor. Early designs were designed to use the 7.62x54r, which was used in the Mosin-Nagant (a full powered bolt action). This early rifle was tested by the Imperial Russian military and considered good, but not good enough. This early rifle had a five round magazine and was heavier than the mosin-nagant. So it didn't make a lot of sense to switch over. Federov also had a hard time lightening the weapon because then it could no longer fire 7.62x54r reliably. The later more successful model which resembles an assault rifle fires 6.5mm a Japanese rifle cartridge. The avtomat chambered in 6.5mm was ordered to be put in production by the Imperial Russian government in 1916, with the plan being that 25,000 would be produced. However the Russian Civil War resulted in only ~3000 being produced. The Soviets concluded that the avtomat was acceptable to be used by the Red Army, but due to supplyconstraints, and not wanting to rely on foreign ammunition the Avtomat was abandoned. It's important to note with this information that 6.5mm is a rifle cartridge, albeit a small one. The Avtomat might better be classified as a battle rifle over an assault rifle.

From a strictly technological pov this is a deceptively subtle question because while most people will agree that an assault rifle is a select-fire rifle firing an intermediate cartridge, concepts like "intermediate cartridge" aren't that easy to define.

One early contender is the Fedorov Avtomat, which initially fired a unique 6.5mm round that had less muzzle energy than the standard issue 7.62mm, certainly fitting one definition of the "intermediate round." The main production rifles were made in 6.5mm Arisaka, which was the standard Japanese rifle cartridge and therefore not intermediate, but very similar to modern intermediate cartridges like the 6.8mm Remington SPC so... intermediate?

Another example is the Ribeyrolle 1918. It was basically an early submachine gun, but firing a powerful (for a submachine gun) proprietary round. So... intermediate?

The Ribeyrolle 8mm round was based on a necked-down .351 Winchester round, a cartridge designed for the Winchester 1907 semiauto carbine. This is only as powerful as a modern magnum pistol round, 1900 Joules, and fires a rather fat, round-nose bullet. Not typical of an "assault rifle" but the French army did buy a bunch of Winchester 1907s, modify them to fire full auto and set them up with extended, high capacity magazines. Is this an assault rifle?

Now I think all of that is kind of irrelevant and I mentioned it only to illustrate that this rivet-counting, feature checklist approach isn't going to lead to a good answer. In my opinion, we should instead talk about doctrine. The assault rifle isn't a gun with a particular set of features, it's a gun made for certain purpose, which is to replace both battle rifles and submachine guns for general front-line infantry, and the first rifle made for this is... the StG 44.

Which is not to say that the modern assault rifle is descended from the StG 44. As you saw above, a lot of different countries were moving towards these ideas even before WW1. Russia/USSR, the British, the French, the Italians and the Americans all had at the very least prototypes of assault rifle type firearms even before the end of WW1.

Follow-up question: why isn't the Browning automatic rifle considered an assault rifle?