As somebody who watches anime with a priority towards a more character-centric appeal, I find that the most interesting characters tend to be the ones I find the least relatable. Characters who's existence challenge me to put in the effort to learn and empathize with them tend to be the most compelling, and having characters who contrast my own beliefs and personality can highlight the changes I could make to better myself.

That isn't to say that relatable characters are uninteresting of course. In fact my favorite character in anime is Tsubomi Hanasaki, one of the most compelling depictions of an introvert I've seen in anime, and the one character that I'd turn to if you asked which anime character is most like myself. I found Tsubomi in particular interesting and endearing because she's not a clear cut "is shy and awkward" kind of character because the show prefers to focus in on her interpersonal relationships and how her differences with others helps her improve as a person.

I would say that the strongest way to provide characterization in anime tends to be providing a cast of characters with contrasting personalities and problems in order to highlight their flaws and room for growth. The characters that you initially relate to, who display a significant sense of growth, are often the ones who best inspire positive change in viewers.

TL;DR: Go watch Heartcatch Precure. Its characters are phenomenal.

Or watch Tatami Galaxy, Hibike Euphonium, or Danna ga Nani. Those have awesome characters too.

"It's just a show, you should take it for what it is " is a valid argument, but I can't seem to conform to this.

I always found it odd when people seem to be under the impression that anime exists purely as an entertainment medium where it's all about bright colors, cute characters, and time passes. More often than not, these people do in fact have an emotional investment in most shows that they watch and love. They just haven't put in the effort to articulate what makes certain shows/characters so deeply personal to them, which is just as understandable as it is unfortunate.