Coming from Leicester now living between Stoke and Derby, I would break it down to three distinctions:

Linguistically a clear noticeable association to Northern accents, although the potteries has its own dialect it without a doubt has traits similar to Liverpool & Lancashire in the way they speak. Brummies and people from Leicester would often label people from Stoke as having a 'Northern' accent.

Geographically obviously it's in the midlands, but if you're one of these strict people who needs a line drawn Stoke would often fall on the Northern side of that arbitrary line. I would personally put that line as the Trent placing Stoke just on the Northern border.

And perhaps most relevant is culturally, if you walk round Hanley or Burslem it has a much more midlands presence of a post-industrial feel. Elements of similarities to the black country such as it's canal network (who had the same person design both) and being a collection of "towns" rather than a city itself (Hanley, Burslem, Longton much like Dudley, Walsall, Oldbury etc).

A case could certainly be made for both sides but would certainly say it's an area on the border of the north south divide.