I don't know if the all coach blankets have tags or not. I would be surprised if Airlines consider coach blankets , which if you are lucky gives ( versus sells) headsets, acceptable to be taken as free flight perks.

In terms of a FA saying it's okay, they are probably taking the path of least resistance. Inasmuch as they don't own the blankets, it isn't theirs to give away.

I do understand when a flight is seriously delayed and passengers are camping on the floor.