I feel like I make this connection a lot more with anime than other mediums, even though it seems like it should be the complete opposite. Anime features are not afraid to take us to all sorts of crazy locations with unique and off-the-wall characters, but that doesn't take away from our ability to connect with the cast. As fantastical as the settings and story can be, at the heart of most anime are characters that you can measure up with.
Take for example, Birdy the Mighty: Decode. This series does a fantastic job of showing you all sorts of amazing characters from far-off planets, but they are tied to something more set in reality by the relationship between Birdy and Tsutomu Senkawa. An earth boy and an alien girl that go through a lot together, and they both learn about each other's worlds through some very common yet deep emotions. It's very hard not to identify with both of these characters in some way.
What anime have you seen where you find yourself really connecting with a character? What anime character can you really relate to, be it through their struggles or background?