HBO sent a DMCA takedown notice to a piece of art that used the phrase "winter is coming".

The article suggests that at some point in this issue, people confused copyright law with trademark law. The two have nothing to do with each other.

HBO has no copyright claim based on that phrase. That phrase might be a trademark, but the DMCA is only concerned with copyright law and is not applicable to trademarks at all. Anyway, using a trademarked word or symbol in a noncommercial way is not trademark infringement.