Learn English – a “pony soldier”

The term retail politics is the opposite of wholesale politics and they are established political terms. The link shows the difference between the two:

Wholesale politics: A mode of campaigning that involves indirect contact with citizens, such as running campaign adds.

Retail politics: A mode of campaigning in which a candidate or campaign staff contact citizens directly, as would happen at a rally, a talk before a small group, or a one-on-one meeting between a candidate and a citizen

As you can see, the original definitions of retail and wholesale are used for politics to differentiate the two. Retail politics focuses more on direct (eye-to-eye) contacts while wholesale politics focuses more on mass media such as radio, television and newspaper advertisements.

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This use of the word "presidential" is using Oxford English Dictionary definition 1.b.

U.S. Having a bearing or demeanour befitting a president; dignified; confident. Also: appropriate to a president; stately; impressive.

The word was originally used to refer to presidential candidates, as opposed to a sitting president like President Trump, when discussing whether their behavior and conduct was of the kind that made them fit to be a president. That is still the way in which it is most commonly used.

During the 2016 U.S. presidential election, there was much discussion about whether candidate Trump had a demeanor that was presidential, or befitting a president.

When asked about his demeanor during the campaign, Trump famously said that if he was elected president:

"I will be so presidential, you will be so bored."

The writer in the Washington Post is insinuating that President Trump's remarks about Charlottesville were not appropriate for the office of President, or in other words not presidential. The writer is contrasting that behavior with what a presidential president would have said, by which they mean, these are the remarks that befit the office of President.

Edit: In your second question, you ask:

Do you say 'vice presidential vice-president, 'chairpersonal chairparson,'directorial director,' 'managerial manager, 'doctoral doctor,''paternal father, maternal mother,' 'childish child,' and likewise?

The reason most of these phrases would be unnatural is because the adjective presidential meaning "befitting a president" is unique. It is an outgrowth of usage.

As with most adjectives that us the -al or -ial suffix, its initial meaning was of or pertaining to the noun, as with OED 1.a.

Of or relating to a president or presidency.

So uses in this sense would be something like "The presidential campaign was underway." Uses like this were extended to the sense cited at the top of this answer gradually over a long period of time. You can see the difference between the meanings in this sentence:

X, who is a presidential [sense 1a] candidate, has an appearance that is very presidential [sense 1b].

When discussing who should be chosen for president, it was and is reasonable to ask, "Does this person have a presidential appearance/demeanor/style?"

However, directorial, managerial, and doctoral never underwent these shifts in usage.

The only sure way to know whether an adjective derived from a noun can be applied to itself this way is to consult the entry for the adjective in a thorough dictionary.