What makes a writer is a question which will have many different answers depending upon who you ask. However, IMHO, a writer is someone who really can't help but write. It is in their genetic makeup.
The question more definitive for me is have you as a writer honed your talent and work so it is publishable for the world? Or do you write for your own pleasure and/or purpose? Either is fine and both have merit.
There have been many writers over the years who refused to publish their works, and yet they had volumes. And many writers whose work should have remained unpublished, but did not. :)
However, if you are a writer, you will know. And your soul will feel release when you write. So be encouraged to write if this be the case.
Happy writing Writers!