Should a teenage girl own a pocket knife?

Why not? A pocket knife is a tool. Pocket knife is a folding knife, of any size, usually smaller though. Such a tool can come in handy in many situations. The one I keep in my pocket is hooked (like a claw), because I find that shape more useful for things like cutting open packages, prying laces loose when a kid has tied them into a knot, careful slicing, and more. A pocket knife should only be sharpened on one side, of course!
My daughter, who is almost 10, has started carrying one around, too. Again it is quite small, with a blade only about 2 inches long. But hers has a couple of accessories on it, a small screwdriver and another one (which I can’t recall what at the moment). While I know she doesn’t use it often, she has told me that it has come in handy for similar reasons. But my kids are very blade aware and can handle knives better than most adults.
If you view at as tool and want to use it as one, learn to use it properly. That makes it a whole lot safer. And it gives you the knowledge to treat it with the respect it’s due. One of my son’s friends recently tried to use a knife and slipped, cutting the tendon through on his thumb. He’ll have a nasty scar to show for it, but it should heal within three months (he’s already been operated on to repair the tendon). He had no training at all in how to use a knife properly. Most people don’t. It’s just trial and error for them. And this is the kind of error that can happen. And that’s why I taught my kids at a young age to use a knife properly. And how did I learn? Decades of trial and error, mostly. But I also have used knives for sculpting and many other practical tasks, so I learned more than the average person. And I also have a couple of scars to show for that learning. Thankfully nothing serious (the worst was a vertical slice down half the length of my thumb, slicing to bone, but no tendons cut, at the age of 16, while sculpting. Yes, I’ve been into art as long as I can remember).
I started teaching my kids as soon as they each showed an interest in learning to cut, and both times it was food at the table. One at age three, the other at age four. They were both cutting their own meat before they went to school.
I found this very basic tutorial on how to use a knife (it’s oriented towards the kitchen), which has some good tips in it: http://extension.usu.edu/files/publications/publication/FSC_Educational_Tools_2011-01pr.pdf
While that’s not everything there is to know about using a knife, the safety part is pretty good. If you know anyone who handles a knife on a regular basis, s/he should be able to give you decent safety and use & care instructions. Be aware that there’s a lot of b.s. on the internet, if you go looking there for information.
A pocket knife should never be viewed as a weapon, as that’s not its purpose. And using it as a weapon is likely to only endanger you more, for a variety of reasons I won’t get into here.

Authentic XYJ Since 1986,Outstanding Ancient Forging,6.7 Inch Full Tang

Should a teenage girl own a pocket knife?
Yes.
I’ve owned a Swiss Army Knife since I was five or six years old. Having such a tool fosters independence and self-reliance, a child learns what can be done with tools. It also fosters discipline, kid cuts herself a few times, and learns to be mindful of blade safety. It also shows that you trust the kid, which builds self confidence.
While it is ultimately up to the judgement of the parents to assess whether their particular child is ready for the responsibility of a pocket knife, I believe as a general principal that a child should be given a Swiss Army Knife, or reasonable facsimile, at the earliest age s/he demonstrates the ability to handle one responsibly.

Chicago Cutlery Belden 15 Piece Premium Kitchen Knife

Definitely!
All the other replies have reasons why covered; However, on a side note, I highly advise you adhere to the laws of your state/country. This includes type of knife, blade length, places where you can carry it, etc. You can always avoid some of these to some extent, but be aware of legal repercussions you may face.
I assume you want this knife for self-defense. If this is the case, I want you to consider your decision making. Knives are dangerous first and foremost, and you need to be prepared to take someone’s life and know when that would be justifiable
This is the knife I own in the state of CA: Amazon.com: SOG Fielder Folding Knife FF38-CP – Mirror Polish 3.3″ Blade, G10 & Stainless Steel Handle: Home Improvement

Smith & Wesson Extreme Ops SWA24S 7.1in S.S. Folding Knife with 3.1in