Gun Control is like trying to reduce drunk driving by making it
tougher for sober people to own cars.
From a UK perspective, where we have tight gun control, "Gun Control" effectively means a system to prevent anyone demonstrating certain indicators which may make them more likely to misuse a gun. These indicators are usually held to be a violent and / or criminal past, any involvement with drugs and certain mental health problems where it is felt the risk of future violence is increased (such as depression).
Applying the same rationale to car ownership would mean preventing anyone where there is a high probability that someone may misuse their car. So, that might also include preventing the legal ownership of a car if you've had a history of drinking and / or substance abuse or a criminal record involving vehicles in some way.
So applying the same precondition could make it more difficult for some (sober) people to own cars. For example, someone who's used a car in a ram-raid (robbery) would not be allowed to own one in the future.
So the analogy is true?
Essentially yes, but it is more accurate to state the analogy as @Tracy Cramer did on her answer:
Restricting citizens with mental illness from owning guns to prevent
gun violence is like restricting citizens with known drinking problems
from owning cars to prevent drunk driving.
Which seems fairly logical...
What the argument is trying to imply is that gun control unfairly restricts responsible, law-abiding people from owning guns. Again, from a UK perspective this isn't really true.
What the argument alludes to is that any restrictions on people that are law-abiding etc. doesn't have any impact on the people that are not. For example, if you stopped sober people from owning cars, but still allowed drunk people to own them, you're unlikely to prevent many incidents of drunk driving.
The gun control argument is trying to highlight that preventing violent criminals from owning guns should be the priority, as they're the ones most likely to commit gun violence.
However, in both cases, whether you have a history of drinking or violence, this doesn't preclude the people who have no prior history from misusing their car / gun in the future. That's another debate entirely.
* To be clear, I'm not siding on one side or another of the debate, just the merits of the argument referred to by the OP **