Po appeared in right place at the right moment. Just when Oogway was choosing the Dragon Warrior, he accidentally pointed at Po (instead of Tigress?). As Oogway later tells Shifu there are no accidents so appearance of Po must be planed/influenced by some other forces.
Shifu: The panda? Master, that panda is not the Dragon Warrior. He wasn't even meant ...
The idea of being split into two can be broken down into different concepts
Good vs Evil
example: Good Ash vs Bad Ash in Army of Darkness.
Here, we have our character who is generally seen as inherently "good". The clone/twin is inherently evil. The evil character is often a "mirror" or complete opposite of the original hero.
He died (probably).
According the the script for Kung Fu Panda, Tai Lung got himself Ka-Thoomed.
Since it happened off-screen, I think we can be reasonably sure that he died, rather than experiencing any mystical conversion to a spiritual plane of existence.
TAI LUNG: You're bluffing. You're bluffing! Shifu didn't teach you that.
PO: Nope. I ...
For those wishing to avoid spoilers, don't play the videos linked here.
There's a few reasons for Kai just not immediately leaving. The end of this video from the very beginning of the movie sets the initial context.
We see two things here.
Next, we have this scene
Later on, we have the ...
"I'm not a big, fat panda... I'm THE big, fat panda!"
This response by Po to Tai Lung's taunts shows that Po has accepted, created, and mastered the fighting style that best suits him, to the point that he typifies that style. With time, his name and fighting style may become synonymous and interchangeable.
It's possible, in the same way, that Tigress, ...
I believe Tai Lung survived in Kung Fu Panda, and is not in the spirit world.
The next time you see him, he had his chi taken by Kai, which would lead one to infer that Kai later fought Tai Lung, with Kai winning and taking Tai Lung's chi.