Many different business models can be monetized - the difficult part is just changing your perspective enough to find the right way.<p>We need some more information to say though. I understand if you don't want to give away what you're doing in detail but a few points about the business would be good (industry, market, etc.)<p>As for it being a startup without being profitable - Yes. Just because you haven't found profit yet doesn't mean you won't.
The question of whether or not you have a startup is determined by your intent. Are you planning on making money once your product achieves scale? Or is it simply a hobby or something you do for fun? Remember, that few businesses(and even fewer tech businesses) are instantly profitable, but if you stick with it, you'll find a business model that works for you.
I've spent a lot of years helping other folks and have not figured out how to monetize it. I would say if it has the potential to be a startup, you need to resolve the monetization issue to get it there. You haven't really given enough info to determine if it has the potential, only enough to indicate to me that you have a defined problem that needs to be resolved if it is ever going to become a 'business' at all, much less a 'startup'.