Being 15 is old enough to have a door. She’s a teenager. At 15, most teens know all the sexual stuff these days. Some of them are also engaged in such activities. Besides it’s an age where she already has her own sexual needs. It does not mean she needs to have a man, but she needs to have privacy at least.
Apart from that, there are things that she might want to keep private. Maybe she is depressed and cries a lot. Maybe she wants to have privacy when she’s talking to her friends. Maybe she has some hobbies that she doesn’t want you to know about. Maybe she has a boyfriend and she does not want anyone to know whenever she’s on call with him. It doesn’t always have to be a “boyfriend”, it could be simple things. Those are things you find out by being friendly with your daughter rather than trying to be nosy. In fact, that’s the right way.
A teenage girl won’t want people to randomly be able to open the door to her room whether she likes it or not. It’s a common behavior among teens where they stay in their room with their doors closed. I did that a lot when guests were around.
I don’t think it’s a problem as long as she is safe, doing the right things and she is happy.
In my teenage years, I hid the fact that I was depressed and suicidal. It’s not the best thing to hide, I know. I had written several journals throughout the years which I didn’t want anyone to see. Ever. I burned them all some time ago. At 15, I was also dating someone. But I didn’t want people to know. So we barely talked over phone calls. I had already experienced myself sexually at 15, but I hadn’t done anything with a guy. My mother educated me well on that topic, so I never engaged because I didn’t think it was worth the risk. I had grown enough to not change clothes in front of my mother, so I definitely wouldn’t be okay with doing that in front of other people.
It’s understandable that she’s a young kid in your eyes, but she’s grown enough to have a door. Respect her needs. Respect the privacy that she wants. Don’t tell her no.