Most likely this refers to the number of gears.
The phrase "standard shift" nowadays refers to the type of transmission: manual transmission or automatic transmission according to which is considered "standard" in a given context (e.g. manual is generally more common in Europe, automatic in the United States). However, automatic transmission was only developed in 1921, one year before the setting of The Great Gatsby, and didn't start being used in cars until the 1930s. From Wikipedia:
The automatic transmission was invented in 1921 by Alfred Horner Munro of Regina, Saskatchewan, Canada [...]. Being a steam engineer, Munro designed his device to use compressed air rather than hydraulic fluid, and so it lacked power and never found commercial application. The first automatic transmission using hydraulic fluid may have been developed in 1932 by two Brazilian engineers, José Braz Araripe and Fernando Lehly Lemos; subsequently the prototype and plans were sold to General Motors who introduced it in the 1940 Oldsmobile as the "Hydra-Matic" transmission.
Clearly this phrase in The Great Gatsby doesn't refer to manual vs automatic transmission, because automatic transmission in cars simply wasn't a thing back then. So what does it mean?
Well, one thing which was changing in car gear systems even before the introduction of manual transmission was the number of different gears. From Wikipedia:
Until the mid-1950s (earlier in Europe and later in the US, on average) vehicles were generally equipped with 3-speed transmissions as standard equipment. 4-speed units began to appear on volume-production models in the 1930s (Europe) and 1950s (USA) and gained popularity in the 1960s; some exotics had 5-speeds.
What this paragraph doesn't mention is that 4-speed transmission was already a thing even in the 1920s on some very high-end luxury cars. In particular, the Rolls-Royce Silver Ghost, said to be "the best car in the world", was manufactured as a 4-speed transmission from 1913 to 1926, while almost all other cars were still 3-speed transmissions. (This is comparable to the modern Porsche 991 starting with 7-speed transmission which most of us have never seen.)
Gatsby's car, although its make isn't specified in the novel, is certainly high-end luxury and quite possibly a Rolls-Royce. Hence the question of whether the car's gearstick is standard shift (3-speed transmission) or the new alternative (4-speed transmission). This fits with Peter Shor's observation in a comment that, in context, the alternative to standard shift seemed to be harder than standard shift.