What do you call an old-fashioned poster pole?

Solution 1:

The proper name appears to be Morris column.

Wikipedia has

Advertising columns or morris columns (French: Colonne Morris, German: Litfaßsäule) are cylindrical outdoor sidewalk structures with a characteristic style that are used for advertising and other purposes. They are common in the city of Berlin, Germany, where the first 100 columns were installed in 1855. Advertising columns were invented by the German printer Ernst Litfaß in 1854. Therefore it is known as Litfaßsäule (Litfaß column) in Germany, and can be found all over the country there.

In France, the columns are named Morris after Gabriel Morris, a printer, who held the concession for advertising in 1868. They were originally built by La Société Fermière des Colonnes Morris. Today, they are mostly built and maintained by the JCDecaux company, which purchased the original company in 1986.

Another source calls them Morris columns and aedicules, which appears to be related to the Latin for "small building".

Solution 2:

I would use the term kiosk advertising. I have bought kiosk space before and that is the terminology that space renters use to charge you.

Just a note too. Looks like if you add the word Victorian in front your results get more exact. I see to it referred to as Victorian advertising kiosk/stand. If you take advertising out then you just get a normal looking kiosk or information center. Looks similar but no ads.

Solution 3:

Advertising pillars but also called Morris columns. And they do come in a variety of shapes and styles.

(Andrew Leach beat me to it, so I'll just post a few images instead.)

Example 1 Example 2 Example 3

On a blog dedicated to Copenhagen

In Absalonsgade (Absolon's Street) is a very old advertising pillar (3) complete with at least 50 year old adds. That is not because the people who service this one have neglected it. This part of the street is part of the Museum Street of the Copenhagen City Museum and they have renovated the street with the original pavement, gutters and street accessories.