I will say that is very rude. However, depending on what the person is accustomed to, it may be more or less rude.
For example, my great-aunts would all talk at the same time, they weren't being rude, they would talk and listen and just have a good time. Then they would ask my mom why she was being so quiet. She prefers to wait until people stop talking to take her turn. Well, my sister and I also tend to talk at the same time, interrupt and talk over each other, we can have a conversation for 3 hours in a very loud, broadcasting voice, we aren't being rude, we aren't angry, we are just enthusiastic and we are listening to the other one even while we talk. So it doesn't bother us a bit, we just are "feisty" and press forward in our conversation. This is our preferred "style." My dad can hold his own with this style. However, my mother (still) and my brother don't like to butt into our nonstop conversation, they want us to just randomly "be quiet" and give them a turn, which seems foreign to us, how are we to know they even have something to say? They expressed their frustration in not having a turn to speak, and we encouraged them to go ahead and speak up more forcefully. So we are still working on it, now when I see then try to start a sentence I consciously have to shut my trap to let them have their say. And this is just the dynamics within family! We've known each other for soooo long and see each other all the time and we're still working on it.
My pet peeve is some acquantances you'll see in public. They come up, super friendly in style, greet you, ask how you are and maybe a few other questions/comments, all without letting you answer, and then their eyes dart over to someone across the room and they run off to do the very same thing to them. To me this is very insincere. Being "friendly" when you never listened for an answer to any one of your direct questions and then darting away is just rude. And they do this to essentially everyone they meet. One girl I know who does that is very nice and well intentioned, but I still find it rude and annoying.
I'm still learning the art of conversation, and it varies with each person you speak with, but I try to 1) speak up in a strong voice and have something to say, especially when addressed. No one likes awkward silence. 2) Listen when they speak. If they are quiet or shy I make an extra effort. If they're a windbag, I don't mind interjecting some.
I'll also add that I have a naturally loud voice and the easiest and most relaxing way for me to talk is loudly. When I'm excited, I can end up essentially yelling, even. It's very difficult for me to talk in a quiet voice and I dislike it immensely. But some people have very quiet speaking voices, my aunt for example is still only medium volume when she is yelling at the top of her lungs, so in other words people talk over her all the time just because she can't compete in volume, her voice is so quiet people often don't know she is speaking. It's also very difficult for me to listen to a quiet talker, it is mentally taxing and requires a lot of concentration, and the whole time I'm thinking JUST SPEAK UP.
But if I were talking to someone like you described... well, I just don't think I would. I'd give up trying to talk, remain silent, and then see if I could get away from them and go talk to pretty much anybody else.
As someone mentioned, some people are very lonely. I apparently have the face that invites perfect strangers to come over and share their very personal story in a random place like a store, and it is "too much information"! It's awkward. I probably wouldn't share some of those details with my close friends if I were them, a lot of times these are people with serious problems, and yet here they are spilling their guts to me. Very awkward, indeed. I can only think they must really be lonely and need someone to talk to.