Is writing the author's name on his books considered showing off, especially with the presence of satisfaction and the expectation of financial benefits? And can it be said to him: "You wrote so that you would be called an author"?

A Muslim must be sincere to Allah in all their affairs and avoid showing off (riya'). Riya' is observing people in one's actions with the aim of gaining benefit, praise, or avoiding harm or blame from them. As for writing the author's name on a book, it does not cause harm.