Stars are born through the fusion of light atoms and the star's nucleus. So let's say that as a star is being born, the nucleus split and creates two stars. Could this even happen? If so, would the stars be considered twins?
This isn't how stars are born. But yes, your premise works.
Here's how you make a star, in a nutshell:
- Take a big cloud of gas and dust.
- Have the cloud grow until it reaches roughly the Jeans length, at which point the cloud is unstable.
- Let some perturbation - perhaps a shock wave from a supernova or radiation from other stars - create a change in density and pressure.
- The cloud should collapse into a protostar, a giant blob of gas that isn't quite the finished product.
- Let the protostar contract via the Kelvin-Helmholtz mechanism until it gets hot enough to fuse hydrogen.
Voila! A star is born.
Binary systems do exist. The leading theories for how they form is that fragmentation happens during the collapse of the cloud, meaning that it splits in two. Each smaller collapsing cloud then forms a star, creating a binary system. Therefore, you can indeed have your "twin stars".
The idea you're talking about resembles the so-called fission hypothesis (see section 2 here), where one protostar, early on, splits in two. There are quite a few problems with the idea, including features of the gas's compressibility. The fragmentation hypothesis is much more commonly accepted.
If you're looking for a more detailed overview of the topic, you may find this paper illuminating.
No, not after the star is born. But this is happening all the time during the star formation - this is called a binary star. However, there is no guarantee that binary system will consist of two twins - typically one star is bigger, and the other is smaller.
Maybe if a white main sequence star was hit by a red dwarf, you might be able to get two orange stars out of the deal, but that's just off the top of my head. I don't have any specific experience looking at these collision in simulations (or papers on the same) to back up that idea (conservation of angular momentum might play a key roles).