I think you're a real writer for writing something. The ones with published work or get paid is not different than the ones that doesn't, in a sense of the meaning. The former just make it their job/profession.
Just like people that draws to their heart liking and not selling their artwork, they're still artisans.

I've always loved your works, please never be discouraged or doubt yourself. :)

A real writer is anyone who writes, that's it. Now, a hobbyist writer is what most of us are. A professional writer sounds like what you want to be. A professional rewrites drafts, gets opinions, needs a marketing plan. However, remember the world is changing, you no longer need a publisher in the traditional sense of the word, and you don't need to distance yourself from fanfiction and your current fans to be successful (ejem, 50 shades of gray)

Hello my lovely, please don't feel discouraged!
Honestly life becomes very difficult when you care too much about what other people think and what other people's definitions for things are.
What matters more, to be a good writer or a real writer?
To write a rubbish story that gets published or post a brilliant story that stays online?
The difference between us and people that get their books published is not that they are 'real writers' and we are not necessarily.
IMO the difference is is that published writers got up and got their works published - that is the only difference.
Even in the published writer's community there are good and bad writers, you know?
It's all about your perspective. If you take out the 'published = real' idea, just imagine there is no such thing as publishing, what is a REAL writer for you?
With regards to using idols who automatically fill up certain aspects of character building, mate, this is something I have struggled with too and I think it has to common amongst the fanfic community. But this is just about doing a bit of studying. Look up how to build original characters, you don't have to build a whole portfolio like some authors do, just gather info that works for you. Being able to build a character can be a really amazing experience, just go slowly if you feel anxious about it. No doubt you have built most of a character already when you write fanfics about idols because no one really knows any of our idols' personalities, no matter how much we think we do.
Having self-esteem in something as personal as writing is difficult and it's a process and journey.
You write really beautifully and eloquently as it is, don't think that it is purposeless.
Why do you write? To feel like a 'real writer'? What does being a 'real writer' feel like? We are all human :) some writers have degrees, some don't. Some study writing in education and some study by themselves because they are passionate. Some writers write beautifully and some don't, some get published and some don't.
This is the most garbled comment I have ever written in my existence but I hope it helps you think a bit and not be discouraged <3
And now I'm off to your account to read your stories because I think you might just be a brilliant writer that I have coincidentally stumbled upon xx

Pffffftttttt you don't need someone else's physical confirmation to be a writer. Publishing a book might make someone a "real" writer, but what constitutes as a "real" story? Everyone who writes have real stories. They are writers as well. Anyone can be a writer. The conventional idea of a writer has been developed because people didn't have fanfiction or any other means to publish their stories other than physical books. Just because your story is based on someone elses's personality or it's published only online doesn't mean it's not real. Write what you love to write. The idols we write about might serve as a basis for our characters but whatever we write will not portray them 100%. We use their names but we can never predict what exactly they will do in each situation. That's where you, the writer, comes in. Because in every story, there is a part of the writer's own personality, own beliefs, and own experiences in it. Your stories are worth it because you put your own time and effort into it. Never believe that just because your stories use a real person's personality or name that your stories are just a cheap copy of them. Fanfiction is just as real as any other novel you can find out there. Some writers depend on idol personality more than others but in the most precious sentimental moments, it's always the writer that comes out to say what they want to say. <3

As far as I am concerned, you become a real writer when you have a deep rooted desire to write. To quote the line from Sister Act 2, "If you wake up in the morning, and you can think about nothing but writing, then you're a writer." Once you publish a book or a story, you are a published writer. When you receive some kind of payment for it, then you are a professional writer. But at all stages, when you want to write, when you have that desire or need to write, you are a real writer.

I used to doubt that my stuff was good enough to publish, but I loved writing. Because of some personal things going on, I have lost that desire to write stories, but I know it's there because I still enjoy writing, even comments like this. Getting my thoughts and ideas into long form makes me feel happy. When I write, I write for me, I write for my enjoyment, because I can put my hand on my heart and say that I love writing. That love makes me a real writer, of that I have no doubt.

As long as you have that love of writing and that desire to write, whatever it is, regardless of whether it is published or not, you are a writer. Whether you are a good writer is subjective and depends on how your readers take your work, but you are no less a real writer than any other published and/or professional writer. You might not be as good as J K Rowling or Tolkien or Stephen King, but nobody else is either. Love writing, and you will feel like a writer.