Father Gonzaga first speaks to the Old Man in Latin and determines that since he (Old Man) does not understand the language of God (Latin), he must be an impostor. Father Gonzaga also uses the slippery logic that since wings are not useful in determining the difference between a hawk and an airplane, they are hardly useful in determining a definitive angel. Gonzaga stubbornly resorts to core church doctrine and does not suppose that he might learn something new about angels. But to be sure, Father Gonzaga writes to a Bishop who will write to another intermediary who will then write to the Pope to get a final answer.
Those at the Vatican wonder if the alleged angel has a navel, if his language is similar to Aramaic, or how many times he could fit on the end of a pin. Some mythologies describe angels as having no navels. But this based on subjective interpretation of angel sightings and myth. It is also anthropocentric (too focused on humanity) to suppose that an angel would speak a human dialect. This connection with Aramaic might go along with church doctrine, but it is arrogant of the priest and the Vatican to suggest that they would know better than God (or the angel himself) how an angel "should" speak. And of course, the notion that the Old Man could fit on the head of a pin is a statement about the immaterial, miraculous abilities of angels. This hardly has any literal or applicable determinations about a real, flesh and blood angel.
In short, the Old Man does not fit the church's preconceived notions about how an angel should appear, sound, and behave. This shows the church's stubborn unwillingness to expand their beliefs. It also shows them to be superficial. Perhaps this is an angel in distress. Wouldn't it be in keeping with Christian generosity to help the angel, rather than to treat him like a circus animal? It seems against church teachings to treat someone as unworthy because of their appearance.
This story is about the subjectivity of interpretation. There is no definite answer that the Old Man is an angel or not. The author leaves it up to the reader. This way of empowering the reader suggests that individuals should be the judges of their own beliefs. The individual should not blindly follow beliefs of a group or institution. The individual should be thoughtful and critical. This supports the idea that the story is a general critique of any body of thought, whether that be organized religion or some other doctrine.