I'm going to have to disagree with Chris.
Any analogy is going to rely - to one extent or another - on the background knowledge of the audience it's being presented to.
In this particular analogy, it relies on the audience knowing what guns are, that there is already legislation in place controlling them, that some people violate this legislation, and that they are proposing more legislation to restrict access to them.
It also relies on the audience knowing what cars are, that there are laws in place specifying who can drive and what is acceptable, and that there are those that violate these laws.
Since Chris did a good job of defining what the argument here actually is I am going to borrow that from him.
Gun control IS TO reducing deaths from gun violence AS making it harder for sober people to own cars IS TO reducing deaths from drunk driving.
However as we can see from the assumed background knowledge of our audience, they know that there are laws in place affecting the use of both cars and guns. Which means that we do have an analogous person on the gun side for our sober person, and heres why.
When we say sober person, we really mean law abiding person since the law specifically states that you must be sober to operate a car. So a law abiding gun owner is, (i.e. someone who does not commit gun violence) in this analogy, equal to a law abiding (sober) car driver.
The next piece from Chris's response I want to tackle is this:
This still has the issue however of the "drunk" stipulation on the right. Again, there's no obvious analog on the left, and again
there's an illegitimate implication (that all gun deaths are results
of the same type of flagrant misuse of guns as drunk driving is a
flagrant misuse of a car)."
The fallacy here again goes back to background knowledge and the fact that the audience knows that gun control is an issue because of the violence for which guns are used.
Which means that it is specifically the "flagrant misuse of guns" that is at issue here.
Since our person being discussed on both sides is a law abiding person, and the issue at hand is the breaking of laws ie: a person committing gun violence, or a person driving drunk,
then accidents must of necessity be taken out of the equation since accidents happen with both cars and guns.
With accidents dismissed from the equation and our background knowledge established what our argument is really trying to say is that:
Restricting law abiding citizens from owning guns to prevent gun violence is like restricting law abiding citizens from owning cars to prevent drunk driving.
Which is an effective statement of absurdity, clearly laying out the anti-gun control point of view. But also a statement that could - with enough research - be verified as true or false. Which to me says that it is a highly effective analogy.
(Interesting note almost as many people die in car accidents as are killed with guns in the US each year, while gun accidents are in most years under 1000 deaths.)
DISCLAIMER: I neither endorse or denounce gun control, that is not what this question is about; it is about the effectiveness of a particular analogy and so my response digs into the reason I think the analogy is either good or bad. NOT wether gun control is good or bad. Also thanks Chris for giving me such a wonderful backdrop from which I could place my arguments against. +1.