Yes, we need to focus on how Higgins as a character changes throughout this excellent play. He is forced to realise that he is not able to treat Eliza as a mere object any more, a scientific experiment. Instead, he comes to realise that he actually harbours feelings for her and that she is an equal not a subordinate character. The quote you have highlighted signifies the way in which he becomes less selfish and is forced to look at others and see that treating others the same is not necessarily possible.
Henry Higgins is, up until almost the very end of Pygmalion, a completely self-assured individual who has never entertained even the slightest momentary doubt about himself, his abilities, or his manner of interacting with his surroundings. He treats every thing and every person exactly the same - taking advantage of their potential to serve the purposes he wishes to accomplish with his life and work, then leaving them without a second thought. Eliza does not understand this because she is accustomed to reacting differently to people from different backgrounds - the rich customer who bought her flowers was addressed in a much different manner than was her father, and Mrs. Pearce is treated in another manner. Higgins does not understand this concept until forced to admit to himself that he has developed feelings for Eliza of a type that he has never experienced when relating to any other human being.
It is really impossible to treat wveryone in the same manner because even when trying to do so one stumbles upon issues related to others' mentality and sensiblity. I think that Higgin's problem is that he wants to change others yet he does not see that he has to change at any level.