To me, it looks like a "Wandering Jew." I haven't had one in many years, but I definitely remember the unique stems. I "Googled" it and found this image, you can see that it has the same segmented typed of stems as your vine. The one is this photo has solid color leaves, but they are most often striped, more similar to yours.
What I recall most about the one I had was that I one time I pinched off a bunch of leaves to encourage new growth and keep it from looking too "leggy." I just tossed the leaves out my back door and the next thing I knew, they had taken root and I had a bunch of new little plants, amazingly easy to propagate! I think you can also start new plants by breaking off pieces and keeping them in a vase of water until roots begin to sprout, then you can plant them.
I hope this was helpful. I'm looking forward to someday having a house with more windows so I can enjoy houseplants again. :)