Well, I cannot say that any anime in particular has really changed my perspective on things, but I would say the Clannad VN, particularly Tomoyo's backstory made me question what a family is exactly.
In the real life, we are told that family is important, but many people get so caught up in defining a family as something that is established by blood (e.g. parent, grandparent, aunt/uncle, sibling, etc.) and/or paper (e.g. marriage certificate, birth certificate, adoption papers), especially because this kind of family gets focused on more in many live-action movies.
However, Tomoyo's backstory showed me that a person can have that kind of family, yet, at the same time, have no family at all.
The only other title that made me really think about things was a manga called Liar Game.
Shinichi Akiyama's talk about doubt, trust, and apathy, and the importance of questioning things help me realize how important it is for me to get to know people and figure out how I can help them, instead of trying to help them in the way society and religious leaders would want me to, actually shows more concern for the individual, and makes me happier that I am who am I am more than any religion could.
Yeah, not quite any anime in here, but many anime seem to be adapted from VNs and manga anyway.