There are several reasons that Donnelly uses the expression “that man” rather than Adam.
The first derives from popular usage. It’s a pejorative construction. Compare “Bob can’t make a sandwich without making a mess of the kitchen” to “That man can’t make a sandwich without making a mess of the kitchen” – omitting the name makes the sentence even more derogatory.
Next, the point of the poem is that Adam, being the first man, stands in for all men. The poem isn’t just about “Adamic naming”, the account in Genesis of Adam naming the animals, but about male control of language and names in general. Think, for example, of how, up until fairly recently, most biologists have been men, and thus all the scientific names of animals are names given by men.
Thus in using “man” rather than “Adam” the poet portrays all of our language as masculine, even the language of the Bible itself which was written by men, and in Irish Roman Catholicism, read aloud in Latin in church by men.