Unusual in modern orthography, and I have only seen it in the New Yorker house style (the diaeresis indicates that the second vowel should be pronounced separately, which is quite different to how that mark is used in a number of other languages that use the Latin alphabet, like German and Turkish). This is subjective, but it reads as a little pretentious to me as a native speaker, compared with other ways of marking it.

More usual is 'co-ordinating' or just 'coordinating' without marking the syllable break between the prefix and the root.

It's part of the New Yorker's writing style.

It is somewhat archaic.

https://en.wikipedia.org/wiki/Diaeresis_(diacritic)#English

https://www.newyorker.com/culture/culture-desk/the-curse-of-...

It’s not a typo, it’s a diaeresis, to indicate the second vowel is pronounced separately from the first vowel instead of being one sound. It’s a typographic convention that’s going out of style on the web.

The New Yorker likes to add the diaeresis in words like this--it's just their style. They do the same with reëlection and other double-vowel words where the second vowel takes a different sound.