The MP-44, later Stg once Hitler changed his mind, was not the first firearm to see military adoption or usage that fit the criteria of an assault rifle. There were guns that came before it that were shoulder fired, had selection options for both single shot and automatic and made use of an intermediate cartridge fed from a reasonable size box magazine. However we commonly refer to the Stg as the first since it really took the concept from the realm of small-time player in to the big time limelight.
It was by no means a perfect weapon and by comparison to the AR and AK of the modern era it has a fair few problems, especially when it comes to the magazines, the heating of the handguard, the absurd design of the fire control group and the lax headspacing of the tilting bolt locking system. That’s without getting in to the manufacturing issues Germany faced during the war. But there’s no denying the practicality of the concept for use at real world combat distances.