If there are two works with the same character name and family type can it be considered a plagiarized work even though the plot is different? One work is about two brothers raised by their father to be supernatural hunters trying to find their mothers killer. the other is about two brothers raised by their parents to be killers for hire trying to survive their abusive dysfunctional marriage.
Names of characters are typically protected by copyright. (I say typically because some very generic names used in many works, like "John Doe", are not attributable to any single original work).
So yes, it would be plagiarism. I cannot create a character named "Harry Potter" and write some other kind of story about him. The fact that you make your "family type" the same would just help to prove you really were plagiarizing that particular character.
I presume you are trying to piggy-back on the fame of some particular famous character in order create "automatic" interest for your work, but that is profiting off of the original author's imagination and marketing money spent, precisely what the copyright law is intended to protect. JK Rowling (and her assigned business partners) are the ONLY people allowed to profit from the name "Harry Potter". (Similarly for whatever name you intended to steal.)
Use your own imagination, and come up with your own fictional name.