Why does the narrator think the Spanish word tener exito is ironic

Because the narrator wants the reader to search what the word means and it means "to be successful" at something!

i just gave you the answer for what the spanish word "tener exito" means, its underlined, in bold and italics!

The direct translation of this idiom is tener - to have and exito - which means success. So this translates 'to have success'. In English we usually say 'to be successfull' rather than 'to have' or 'to own success'.