The writer is selling his own book.
look and compare
Yes, of course! Especially those who publish on their own. Are you a writer?
Should 'The writer is selling their book' not also be acceptable? The swój/swoja/swoje thing only changes with the gender of the object right, not the subject as well?
I agree that "their" is pretty normal everyday use English in this case.
'they' and 'their' are very often used these days as non-gender-specific singular pronouns. I would say this is now very much the preferred form. A few people still write 'he' and 'him' when the gender is unknown, but this is widely considered to be undesirable. More recently still, it has become quite common to use 'they' and 'their' as singular pronouns even when the gender is known.
I understand that, I use it myself in English, but still, using "their" doesn't guarantee that you understand that "pisarz" is definitely a guy.
Oh wait, I absolutely don't understand why you would use "their" when the gender is known, but nevermind my opinion, we will keep to "his" to be sure the learners understand everything about the sentence.