How can an assailant who stabbed the victim be pressing charges on the victim?

It's going to be very fact intensive.  Sometimes the person who gets stabbed was the aggressor and started it, and was stabbed in self-defense.  That's just one way.  The real issue is that the police don't always talk to all parties.  Many times they have tunnel vision about what they want to happen or makes sense to them.  They only need probable cause to arrest, which is a very low standard.  Higher a lawyer to properly investigate this case and be able to show the person stabbed was truly the victim.  

In Texas, police can arrest on probable cause, which is a low bar. When officers arrive at a fight and everyone tells a different story, they can charge more than one person from the same incident, and a person who was hurt can still be charged. "Pressing charges" is also not really up to the victim or the other side. Police and the prosecutor decide, and a prosecutor can dismiss a charge after looking at the evidence.

Self-defense is considered, but usually later. Texas law lets a person use force, and in some cases deadly force, to protect himself from unlawful force. It generally does not help someone who started the fight, and timing matters. A fight at a bar and a second confrontation hours later can be treated very differently. Video, witness statements, and hospital records tend to decide these cases.

The man who was stabbed can also report the people who came at him. A knife attack can be charged as aggravated assault. A criminal defense lawyer can find out how the case was filed, what the evidence shows, and how to put the self-defense facts in front of the prosecutor early.