What does "period charm" mean?

It can be surprising to see it used in this way for the first time. "Period" can just mean some period of time in the past without reference to a specific one. The dictionaries I've checked all list this use.

adj.
Of, belonging to, or representing a certain historical age or time: a period piece; period furniture.
American Heritage Dictionary

of, relating to, or representing a particular historical period
Merriam-Webster

adjective
attributive
Belonging to or characteristic of a past historical time, especially in style or design.
‘an attractive and beautifully modernized period house’
‘a splendid selection of period furniture’
Oxford dictionaries (at Lexico.com)

2.
a. a portion of time specified in some way: the Arthurian period; Picasso's blue period.
b. (as modifier): period costume.
Collins English Dictionary

You can see when used in this way, ie., "period house", "period furniture", "period film/drama", it may not refer to a specific period, it just means a historical period, not so recent.

Does period charm usually refer to some specific "period"? (Must it?) Or can it refer to any period?

Sometimes the period is specified when using "period" in this way, other times it's not. In your example it mentions 1914. In other examples, like those shown in the dictionary there's no such specificity.

According to Google NGram Viewer there are some differences in frequency in BrE and AmE books. "Period furniture" seems to have grown in frequency in AmE in the 70s and in BrE in the 80s. There is a very noticeable spike in BrE for the use of the term "period drama" at about 1990.

Google Ngram chart (BrE and AmE)

My guess is that this use of "period" comes from the arts.

Period drama

A historical period drama (also historical drama, period drama, costume drama, and period piece) is a work set in a past time period, usually used in the context of film and television.
Historical period drama (Wikipedia)

Period piece

n
(Art Terms) an object, a piece of music, a play, etc, valued for its quality of evoking a particular historical period: often one regarded as of little except historical interest
Collins English Dictionary