At the end of the game, right before the party battles Kefka, Kefka asks why people rebuild things they know will be destroyed, and why they cling to life when they know they can't live forever. Terra's response is one of my favorite quotes in the entire game. It should be put on bookmarks and refrigerator magnets.
It's not the net result of one's life that is important. It's the day-to-day concerns, the personal victories, and the celebration of life...and love! It's enough if people are able to experience the joy that each day can bring!
Kefka then asked if they had found their joy in their nearly-dead world, and each character proceeded to tell him the reasons they kept fighting and living. Sabin's is pretty sweet.
I have come to experience anew the love of my brother!
I think this quote says a lot about Sabin's motivations for well, everything. By this point in the game, Sabin knows about the double-headed coin (pending you had Edgar and Sabin in your party aboard the airship after the Opera), and he knows the lengths Edgar went to make him happy. I think that definitely played a part in Sabin's caring for Gau and Cyan, and it was his reason to continue fighting. Edgar cared so much about Sabin, and in caring for Gau and Cyan, he was essentially just doing what Edgar did for him.
Edgar worried about making their father proud, but I think Sabin worried about making Edgar proud. Edgar was the greatest person in Sabin's life, and I think that in a lot of ways Sabin strove to be more like Edgar. More responsible. More selfless. It would certainly explain the dramatic change from his seventeen year old self where he just thought about himself and his own freedom, to his twenty-seven year old self where he cared so much about others even more than himself (like when he jumped in to attack Ultros alone to save the others, not caring about his own safety).
I do think Sabin's father's last wishes had a large part in Sabin feeling responsible for Figaro, but I think Edgar helped motivate him a lot. Edgar was the one that reminded him it was their dad's desire for them to take care of Figaro. While Sabin was less responsible and less mature when he was seventeen years old, Edgar had always been responsible, and took care of Sabin even at that young age. It would make sense that Edgar would be his role model, and that the qualities of Edgar Sabin liked would end up being Sabin's strong points.
I think that Edgar saw how great of a person Sabin had become, and did love and respect him. In the end, I think that is all that Sabin wanted.