From the OQ:
>To me, "I cannot afford this car" means "I am not able to have enough money to be able to buy this car".

The reason *can* is used with *afford* is because people use the word *afford* as a substitute for the word *[provision](http://www.merriam-webster.com/dictionary/provision)* or *provide for* or *allocate/furnish funds for/to*.

The common usage, per [Wiktionary](http://en.wiktionary.org/wiki/afford):
> To give, grant, or confer, with a remoter reference to its being the natural result; to provide; to furnish.


Per the original quoted definition, afford is defined as [already] having the money to be able to pay for something. But people don't use that definition when they say "afford". That is to say, English usage of afford in this context is about *furnishing* of funds rather than *having* the funds. 

"Can you afford to pay for that new widget?" Means: Are you able to furnish the funds to pay for the cost of the widget?

Yes, *to afford* means in one definition to have the money to be able to pay for something, but it's not just about *having* the money (to an English speaker). It's also whether the money *can* be [furnished](http://en.wiktionary.org/wiki/furnish) to pay for it.

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Per questions:

> Why not *I **do** afford this car*?

Because *do* [verb] is almost always used with thinking and feeling verbs, especially for emphasis:

> I *do* like you.  
I *do* like to see this movie.  
I *do* think you're going to like this.

You're not going to *do* afford this car. You either can afford/furnish funds for the car or you pay for the car.

>Q: *Do* you afford this car?  
A: (Do I afford this car [what]?)

vs

>Q: *Can* you afford this car?  
A: Yes

*I **do** afford this car* is incorrect simply because the statement isn't finished. That is to say, the sentence needs an explicit object that is being afforded *to* the car.

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Rewording for *I **don't** afford this car*: no difference in the summary statement. The sentence needs an explicit object that is [not] being afforded *to* the car.