I’ve just done a quick search and it would seem that this exam took place at an earlier age and level than a university degree:
” The title of Associate in Arts, introduced by the University of Oxford in 1857 and sometimes referred to as the degree of Associate in Arts, predates the Durham degree. However, it was an examination for "those who are not members of the university" and who were under the age of 18; as such it was at the level of a high school qualification rather than a modern associate degree. Examinations were held in English, languages, mathematics, science, drawing and music, with the title being conferred on those who students who passed any two (as long as the two were not drawing and music)”
Further information from Oxford University:
None of which detracts from how wonderful it is to learn about your relative’s document, and it must have been an unusual achievement at that time.
Ruth Silman, England