I think you're fantastic, and just what the police as an organisation need. You're not typical (certainly don't seem to be) and thank goodness for that.
When I started at Uni I was convinced there'd been a mistake and I spent the whole of my first year waiting for the tap on the shoulder and the words "Sorry, Jux, we didn't mean you, we meant a different Jux, off you go now". This, despite the admissions officer on our first day, telling us that not one of us was there by mistake; that he didn't make mistakes; that every single one of us was there because we were entitled to be there, had passed all the tests required and were wanted there.
I finally admitted it to some friends sometime in my 3rd year. Every single one of them had spent their first year thinking they weren't really meant to be there either! These were confident, obviously highly intelligent people. We were all in our 30s.
The point is, that you may feel you aren't up to it, but no one else thinks that. You'd soon know about it if they did. And you have no way of knowing whether they don't all feel that they are lacking and not up to it, either.
I think that just by doing the job, you will prove to yourself that you're fit for it. What you may need to decide (a bit further down the line) is whether it is fit for you, whether you find it satisfying and challenging enough for you, whether it is good enough for you, whether you want to do it or want to do something else.
That all comes later, though. In the meantime, continue to do your job, continue to show yourself that you can do your job; tell yourself that most likely a lot of the others feel the same as you. They may put on hard fronts, but I bet loads of them are jelly underneath.