Is this something that gets lost in translation? The version I read came with the original Latin on the other side, and in it, Trajan uses mi Secunde carissime, which translates to much the same thing if my four years of high school Latin serves me correctly.
Was there a deeper meaning to this form of address? Was it casual, in the way that we in the modern day might begin letters with "Dear so-and-so?" Did Trajan refer to all of his higher-ranking servants this way?
It is a highly frequent vocative in Latin letters. You are on the right track with the comparison to "dear so-and-so" in English. I found, for example, this article which, on page 502, is able to give real numbers; this is the third most frequent vocative after dominus and frater in the Vindolanda tablets, and together the three vocatives make up 89% of all Vindolanda vocatives. The article also points out the connection to the exactly equivalent common use of philtate in Greek. Quoting the article:
Writers of papyrus letters use the term [philtate] freely as a mildly polite way of greeting colleagues, associates, and officials, just as carissime is used at Vindolandia.