According to Catholicism, why did Jesus not name the rich man in the story of Lazarus and the rich man?
There is no reason in particular, according to the Church, why Our Lord did not name the rich man in the parable about the rich man and Lazarus.
Or is there?
It is interesting to note that neither Jesus nor any of the popes (as the Vicars of Christ) have ever named someone as being in hell. They have not even pronounced on the question as to the demise of the Apostle Judas Iscariot Was and we all know the grave words that Jesus spoke about him shortly before the Passion: "Better for this man that he had never been born" (Mark 14:21).
Michelangelo (1475-1564) in his Last Judgment placed the Papal Master of Ceremonies, Baigio da Cesene (1463-1544) in hell. When he complained about his image being in hell, it is widely believed that Pope Paul III (1534-1549) responded: "That his jurisdiction did not extend to hell and the portrait would remain" as it is!
The Church in the Middle Age, at least in some countries, did give a name to the rich man of this very parable. According to legend and/or tradition his name was Dives. For the Church in some regions in the Middle Ages, this is an actual event because the poor man in the story is named: Lazarus.
As a literal historical event
Some Christians view the story of Lazarus and the Rich Man, not as a parable, but as an actual event which was related by Jesus to his followers. This was generally the view of the medieval Church.
Supporters of this view point to a key detail in the story: the use of a personal name (Lazarus) not found in any other parable. By contrast, in all of the other parables Jesus refers to a central character by a description, such as "a certain man", "a sower", and so forth. - Rich man and Lazarus (Wikipedia)
This is the only parable that gives a proper name to someone, it is simply that a story told by Jesus to express certain truths. The other parables do not name any participants by name, including the parables of the Ten Virgins, Prodigal Son, and Good Samaritan. It is also the only parable with someone in hell. Possibly these two elements may hold the secret to the question. , according to Catholic tradition.