Even the best writers have doubts about their work. I recently listened to a reading of Dear Elizabeth, which is based on the correspondence between two of America's great poets, Elizabeth Bishop and Robert Lowell. Both of them wondered about the quality of their work, even after they had achieved great success. Unless it becomes paralyzing, I think that's actually a good thing. It can be a spur to do better work and to avoid complacency (something I've seen in later works of writers of great potential).
Of course, the kind of doubt I see most often is among people who have had, so far, limited success. Much of this is based on commerce — works sold, money earned, and awards that really only come to people who either have, or show potential to have, large, paying audiences/readerships. There are some writers, of course, who fret about not having fame or an academic reputation or good reviews or well-known honors supposedly reserved for works of the highest quality.
Doubts related to money can lead to the sensible choice of getting or continuing in a day job. If you have bills to pay, and especially if you have dependents, working is noble, even if it cuts into writing time. A life of poverty can also make it harder to be prolific and create your best work.
As to reputation, unless you're as good at publicizing yourself as Walt Whitman was, it's largely out of your control. While you can continue to improve your work and focus on projects that both inspire you and reach the audiences you choose, a lot is based on luck. (Provided you don't actively offend people who have influence, something I've seen some writers do.)
So where does that leave us as far as doubt is concerned? If you have worries about quality, education and practice provide the answers. You keep writing in challenging yourself. Doubt becomes a positive. If you have financial concerns, you may relegate writing to shorter periods of time. And you treat that time as precious. Even 15 minutes a day, used purposefully with a focus on completing works, will be enough to seriously pursue writing (based on what I've seen with my students). Most writers, including some who are well-known, have day jobs. So doubts about making a living as a writer are well warranted. Having to write part-time maybe disappointing, but it's not fatal to the work. It can lead to creative approaches (such as building connections within writing communities and fashioning a portfolio of works in case opportunity knocks). Or, doubt can be resolved, just by accepting a typical writer's life — one that's split.
What about "am I really a writer" doubts? Here's the answer: You get to decide. If you write 15 minutes a day (even five days a week) and complete works, no one can say that you are not a writer (except you). No one can say you're not contributing when you devote an hour and a quarter every week to something that matters to you.
That last is the key. The work has to be meaningful for you. Not every day. Not every part of it. As an analogy, I like to think about one of my other roles, cat owner (or servant). I love animals, and felines fit into my lifestyle. They are companionship and therapy and entertainment. They also are hairballs and litter boxes and midnight howling. They have pluses and minuses – as does writing. If I were to make a list of the reasons why I devote hours of my life to cats, the list would be long and full of reasons that are important to me. And the same would be true for writing. I strongly suspect that many people who doubt whether they are writers or not put too much focus on the minuses and not enough on the pluses.
If you truly worry about whether you are a writer or not, take the time to list the pluses and minuses. If writings litter box problems feel overwhelming, maybe it's time to quit. Maybe something else deserves your attention. But chances are that most of you reading this will discover or rediscover the reasons why writing matters to you.
Ray Bradbury wrote every day for most of his life. When he had a stroke and lost many of his abilities, one of the first signs of healing was his finding a way to dictate stories. He loved writing. It was part of who he was. Without a doubt.