This is not unique to the USA, e.g. in the Netherlands the army assists in testing and vaccination campaigns [1]. This is just common sense, those organisations have a large number of people (who in peacetime have not much more to do than exercise and maintain their equipment) who are trained to act in a short times notice and can deploy facilities which are not available to civil authorities.

https://www.defensie.nl/onderwerpen/coronavirus-covid-19