Keep in mind the differences between the two guns were a bit more nuanced than how stable they were in full-auto fire - the Sturmgewehr was a heavier and bulkier gun than the AK. Details of performance aside, the reality is more likely due to the massive proliferation of the AK.

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The STG-44, for all its significance, was a late-war German design that didn't see much production. Upper end estimates give a little under half a million examples produced, and their use was limited largely to WW2 Nazi Germany. Postwar, it would see limited use in East Germany, Yugoslavia, and France, but even the most intensive postwar use - by the French in Indochina - took place in lower-intensity conflicts that received little attention in the popular consciousness. French examples largely ended up in former French colonies, which again see little representation in English-speaking culture and thus end up easily forgotten.

The AK-47 and its derivatives, on the other hand, saw proliferation on a scale beyond that of nearly every other weapon. Being the primary infantry rifle of the Soviet Union for nearly the entirety of the Cold War would be more than enough to ensure its place in the popular consciousness, but it would proliferate far beyond that, becoming the standard infantry arm of most Warsaw Pact states (Czechoslovakia being the exception) as well as non-Soviet aligned communist states like Yugoslavia and China. On top of that, we have the fact that AK pattern rifles were regularly provided as aid to any number of proxy conflicts in which the Soviets were involved. In US popular culture, the heavy involvement of Soviet and Communist Chinese aid in the Vietnam War meant that the AK-47 would be heavily associated with North Vietnam and the Viet Cong, but all over the world we see the AK-47 proliferate. In the Middle East, Arab states leveraged Soviet aid for quite some time - Nasser's Egypt, Saddam's Iraq, and Assad's Syria being prime examples - so the AK-47 that was so readily provided ended up being a common sight in the armies of these nations. Similarly, Soviet involvement with both Soviet-aligned states and non-state actors in Africa would see lots of arms flowing into these areas. Ethiopia, the Congo, Angola, and Mozambique all saw heavy Soviet involvement through the Cold War. Given how visible infantry rifles are compared to other, arguable more effective military aid such as training, logistics, and heavy weapons, it's no surprise that the AK-47 became an iconic symbol of these lower-intensity and often asymmetric conflicts. Mozambique even went as far as placing the silhouette of the gun on their flag - a testament to the impact it had on the national consciousness during their war for independence and subsequent civil war.

Like the AR-15 and its derivatives, the AK hit a level of proliferation that would see it start to transcend involvement of its original producers - the Soviets and their allies - in where it showed up. Where states failed to meet their goals during the Cold War, AKs ended up provided to non-state actors in hopes of furthering political goals. While Soviet-backed non-state actors made up many of these groups (think early MPLA, FRELIMO, the Viet Cong), the failures of the Arab states to achieve their goals during the Arab Israeli Wars would see AKs end up in the hands of non-state actors aligned with similar anti-Israel goals. Looking further into the future, the fragmenting of Yugoslavia and subsequent chaos saw the former state's large arsenals fall into the hands of non-state actors. Given that AK-derived rifles and SKSs were the bulk of the Yugoslav arsenals, it's no surprise that they'd end up heavily associated with the imagery of the war. Following the end of the Yugoslav wars, those same arms ended up in the hands of criminal groups that proliferated all over Europe.

In the hands of states, the AK even had the distinction of catching the attention of states patently opposed to both the Soviets and Communism. Finland, who walked a fine line of neutrality between East and West, opted to adopt an AK-derived rifle - the RK.62 Valmet. Israel, finding their Belgian-designed FAL rifles to be poorly suited for the mechanized desert warfare their army was fighting, found a lot to be admired in the large stockpiles of captured AK-47s they acquired during their wars with the Arab states, and they ended up developing the Galil rifle after some testing. The Galil was derived from the Finnish RK.62, and, while it has some notable differences in the design, the core of the gun was an AK-47. The design jumped a step further, with the Galil coming to be adopted by many Central and South American states and Apartheid South Africa.

So when it really boils down to it, the AK's prominence in the popular consciousness compared to the Sturmgewehr is entirely deserved - while one was a relatively obscure gun that saw limited use at the end of WW2 and a few postwar conflicts, the other was the centerpiece of the army of one of the two Superpowers that saw proliferation all over the world on both sides of the Cold War.