This is an answer in two parts, first looking at sociological reasons for left wing support of a restricted market; but I conclude with some thoughts that protecting the taxi industry is actually cross-paritisan.
Is there a sociological argument to be made to explain why the left wing would tend to oppose laws that would permit competition with traditional taxicab operations?
Let me start with a mention of "Moral foundations theory" that was
first proposed by the psychologists Jonathan Haidt, Craig Joseph and Jesse Graham, building on the work of cultural anthropologist Richard Shweder; and subsequently developed by a diverse group of collaborators, and popularized in Haidt's book The Righteous Mind.
The original theory proposed five foundations: Care/Harm, Fairness/Cheating, Loyalty/Betrayal, Authority/Subversion, and Sanctity/Degradation. It now includes a sixth parameter, Liberty/Oppression; while its authors remain open to the addition, subtraction or modification of the set of foundations.2
(This is not without its criticisms, but you asked for "a sociological argument")
What I wanted to highlight is this paragraph summary on the relation to political beliefs
Researchers have found that people's sensitivities to the five/six moral foundations correlate with their political ideologies. Using the Moral Foundations Questionnaire, Haidt and Graham found that libertarians are most sensitive to the proposed Liberty foundation, liberals are most sensitive to the Care and Fairness foundations, while conservatives are equally sensitive to all five/six foundations. Joshua Greene argued however that liberals tend to emphasise the Care, Fairness and Liberty dimensions; conservatives the Loyalty, Authority and Sanctity dimensions.
I want to now focus on this "Care and Fairness" aspect as it relates back to the original question ("Is there a sociological argument to be made to explain why the left wing would tend to oppose laws that would permit competition with traditional taxicab operations?")
Let's take a look back to New York in the 30's before major regulation:
In the 1930s, the growth in unemployment and unsold automobiles produced a drastic increase in the number of taxicabs.10 While fewer people could afford to ride a taxi, the number of taxi cabs skyrocketed, while occupancy rates and revenue per taxi declined.11 Capacity and demand were moving in opposite directions.
An editorial published by the Washington Post in January 1933 illustrates the public's perception of the chaotic state in which the taxi cabindustry found itself:
Cut-throat competition in a business of this kind always produces chaos. Drivers are working as long as sixteen hours per day, in their desperate efforts to eke out a living. Cabs are allowed to go unrepaired....
Together with the rise in the accident rate there has been a sharp decline in the financial responsibility of taxicab operators. Too frequently the victims of taxicab accidents must bear the loss because the operator has no re-sources of his own and no liability insurance. There is no excuse for a city exposing its people to such dangers.12
Economists of the era argued that taxis were a declining cost industry; excessive competition between numerous small operators decreased carrier efficiency and increased consumer costs.13 The U.S. Department of Transportation also summarized the tenor of the times:
The excess supply of taxis led to fare wars, extortion, and a lack of insurance and financial responsibility among operators and drivers. Public officials and the press in cities across the country cried out for public control over the taxi industry.
The response was municipal control over fares, licenses, insurance and other aspects of taxi service.14 
To summarize the above: too many taxis, means drivers are working too long, and because there are so many drivers any one driver can't make as much, and some can't even afford insurance. There's at least a few "care and fairness" complaints -- drivers, through no fault of their own, make less than they did several years ago; or a regular citizen has to incur additional costs of an accident when getting in an accident with an uninsured taxi driver.
Paradoxically, deregulation (now some decades later) also resulted in higher taxi fares:
One would expect that excess capacity would drive prices down, as it allegedly has, for example, in the deregulated airline industry.187 Paradoxically, precisely the opposite has occurred in the deregulated taxi industry. As Price Waterhouse observed, "prices rose following taxi deregulation in every documented case."'188
Professor Roger Teal of the University of California studied pricing at nine cities which deregulated (i.e., Fresno, Kansas City, Oakland,Phoenix, Sacramento, San Diego, Seattle, Tacoma, and Tucson). He concluded, "In every city in this study taxi rates are now higher in real terms than before deregulation, often by a substantial amount."'189 Before de-regulation, in none of these cities did rates rise as rapidly as the Consumer Price Index [CPI]; after deregulation, price increases exceeded the CPI in each of these cities.190 Professor Teal concludes, "taxi rates may have increased as much as 10 per cent more in the deregulated cities than they would have done under continued regulation.' 
The whole article  is good, but the key thing to note is from the abstract:
Beginning a half century later, more than 20 cities, most located in the Sunbelt, totally or partially deregulated their taxi companies. However, the experience with taxicab deregulation was so profoundly unsatisfactory that virtually every city that embraced it has since jettisoned it in favor of resumed economic regulation.
And the summary list on page 102
The experiences of these cities reveal that taxicab deregulation resulted in:
- A significant increase in new entry;
- A decline in operational efficiency and productivity;
- An increase in highway congestion, energy consumption and environmental pollution;
- An increase in rates;
- A decline in driver income;
- A deterioration in service; and
- Little or no improvement in administrative costs.
Some 40-60 years ago cities experimented with deregulating the taxi industry, but this resulted in widespread inefficiencies -- for both passengers and drivers. Since then, the majority of cities have reverted to a protected industry.
Does that mean protection is left wing? Arguably, deregulation was a right-wing proposal to let the free market decide, at whatever cost to existing businesses (including solo driver-owners). However, it seems the results from deregulation were so bad that the majority of people (in a cross partisan manner) agreed it was a bad idea but for different reasons -- on the left (perhaps) that drivers and passengers were harmed, and on the right (perhaps) that businesses (including solo driver-owners) where harmed.