Yes you can bring a blanket onto the plane with you.
A blanket is not considered part of your personal items or carry on items when boarding a plane and going through security.
You can just carry the blanket with you under your arm and they will just see it like a jacket being carried under your arm.
I always just bring the blanket with me in my arms and never had any issues.
I cannot go on flights without having a blanket because I like to put a blanket over myself when I sleep which I do fall asleep on flights especially long flights.