You asked "or is this fundamentally just a marketing success?" I think the answer to these sorts of questions always has to take into account the historical background.
The Hammond organ came on the market in 1935. It became distinctive because it came first. It was popular and sold in large numbers. It was the first commercially successful electronic musical instrument that could be played with a keyboard, and was polyphonic (it can play chords rather than single notes). It became strongly associated with certain types of music due to its popularity and lack of competition in its early decades. In addition, it's also a unique-sounding and wonderfully versatile instrument, but for this answer I will concentrate only on the historical context.
The Hammond was originally marketed as a less-expensive and more practical substitute for an acoustic pipe organ of the sort used in churches. Acoustic pipe organs were tremendously expensive to build and maintain, and typically were a fixed installation; the typical acoustic pipe organ was not portable. Soon there were smaller versions of the Hammond organ that were suitable for use in homes, where a pipe organ would be tremendously impractical. Hammonds could also be made portable, so they could be used in many contexts where an acousic pipe organ could not.
In 1935, the Hammond company could not conceive that their instrument would ever be used in jazz or blues, let alone rock. By the same token, in 1935, organs had never been used in jazz or blues.
Soon the Hammond organ was being used for newly emerging forms of music. Its sustained sound added a new dimension to jazz and blues. Before the Hammond organ, jazz and blues bands could not use acoustic pipe organs because virtually all pipe organs were installed in churches and orchestra concert halls, places that jazz and blues bands were not able to rehearse or play.
At the same time, in African-American churches, the Hammond organ was being adapted to the newly emerging style of sacred Christian music called black gospel. The setting for this was often a church whose congregation was not wealthy enough to purchase an acoustic pipe organ, but could afford a new Hammond. Musical innovation in this environment was such that they adapted and created a new style of music to play on this new instrument.
Then jazz clubs and entertainment establishments began purchasing Hammond organs for use by various bands that would come and play there, in the same fashion as the club would own a piano that all visiting bands would play.
It also helped that a highly-skilled Hammond player could play bass lines with his feet on the bass pedals, in the same manner as the classical church organ, obviating the need for another musician who played bass. So this lowered costs for certain jazz combos.
The Hammond organ is a wonderful and versatile musical instrument, but it became standard because it appeared on the market at the right time when it was a unique type of instrument with virtually no competition among other instrument makers (The Allen electronic organ company, emerging at almost the same time, chose to make only very large console organs that were permanently installed in churches, so they weren't really competing with the Hammond instruments). When the Hammond became commonplace, then it got adapted to jazz and blues and gospel, and ultimately rock.
The Hammond organ was therefore firmly established and very popular long before any competing instruments appeared. Its sound was firmly entrenched in blues, jazz and gospel, and keyboardists who played those styles of music had become adept at getting great sounds and performances from a Hammond instrument, starting in the 1930s and going through the 1950s. Then came the advent of other new keyboards such as electric pianos in the 1950s, transistor organs in the 1960s, and polyphonic synthesizers in the late 70s and into the 80s.
The Hammond organ (or more often, emulations of its sounds using newer technology) remains the distinctive instrument for certain styles of music to this day, even though the original electro-mechanical rotary-tone-wheel Hammond organs ceased to be manufactured altogether in 1974.