Sources: bar magnet (annotated with a red "X"), aluminum can (annotated with a green "OK" check mark)
Beautiful video and demonstration, thank you!
That magnet in microgravity doesn't actually orient itself at all
I watched the video carefully and at
00:44 the first time the magnet is released it is pretty far away from the aligned direction and it does not "immediately orients itself to the magnetic field of the earth." Instead it oscillates dramatically about that point as energy is exchanged between kinetic and potential.
Eventually eddy current losses and viscous drag losses in the air would damp it down (loosing it's energy by heating the magnet and the air) and it would then look like a 3D magnetic compass.
Only two seconds later at
00:46 game developer and private astronaut Richard Garriott(?) recognizes the oscillations are to large, grabs it, and tries twice more until at
00:54 when they re-release it close to the median point, where it still oscillates at the same frequency but at a lower amplitude.
The ISS is not a bar magnet, it's an aluminum can!
The actual bar magnet that wealthy game developer and amateur science video maker Garriott uses is no model for a mostly-aluminum structure in space.
While as @OrganicMarble points out the station's structure is primarily aluminum, there will be enough ferromagnetic material with some residual magnetization (unless everything sent up is methodically degaussed, which is always a possibility) some permanent magnets inside motors with imperfect return yokes, and of course some loops of current that all together are likely to result in at least a tiny residual magnetic moment for the ISS.
Even a good captain of a wooden ship will (presumably) do both "soft iron" and "hard iron" corrections for the compass every time a substantial change has been made to the metal content of the ship.
But mostly you should think of the ISS as more of a non-magnetic aluminum can than a bar magnet.
The ISS is already spinning quite nicely
It orbits the Earth every 90+ minutes and in order to keep its down-side-down it also rotates about one of its own axes every 90 minutes as well. There's a lot of angular momentum stored there and that probably dominates any interaction between any residual magnetic moment and Earth's field.
It also works very hard to maintain that state.
And if it lost that ability to control its attitude and rotation, atmospheric drag (a very lossy and damping influence) would either forcefully align the ISS to a new equilibrium rotation state with a minimum drag orientation, or cause it to spin and tumble wildly.