Yes, and the decision not to let the police in that night complicated things significantly.
Self-defense is a legal defense, not a shield against arrest. When police respond to a call involving a physical altercation and someone has been injured, they are going to detain the person identified as the aggressor regardless of what actually happened. Sorting out who was defending themselves and who threw the first punch is a question for the courts, not the officers on the scene.
Refusing to let police in created an additional problem. Law enforcement responding to a domestic disturbance or assault call and being denied entry tends to escalate their suspicion rather than reduce it. It also means the only account they had to work from that night came from the neighbor, which almost certainly framed your fiance as the aggressor.
The self-defense argument is absolutely still viable, but it has to be built properly by an attorney. That means gathering evidence of what the neighbor did to provoke the confrontation, any prior history between them, witness statements if anyone else saw what happened, and anything that supports the claim that your fiance acted to protect himself rather than initiate the fight.
The most important thing right now is making sure he says nothing to anyone inside about the details of the incident. Not to other inmates, not to staff, and not on recorded phone calls. Everything he says can end up in front of a prosecutor. Let the attorney do the talking from here.