As far as I am concerned, the compass is facing along the magnetic field lines on the Earth's surface which in turn points towards the North pole by its design. But for a magnet, the field lines go from north to south on the surface (outside the magnet). Does this mean that Earth's geographic North pole is the magnetic South pole?
It is. The geographic North is close to the Magnetic South. They're not absolutely coincident, and the magnet is slightly moving continuously due to Earth's dynamics. However, the geographic North pole is well stablished.
However, remember that names are just a convention.
We call it North because the compass points there. That's the reason for the name. Then, if you start thinking, yes: that means a magnetic South inside the Earth, but the name was there before.