Two swords have more overall weapon damage (and thus spell damage on your character sheet) than sword and board does.

This is one of the trickier rules in the realm of eso's damage scaling rules and addendums. For myself, I know that i'll have higher weapon/spell damage with dual wield than I will with sword and board but I'm not entirely sure the best way to explain why this is the case. If anyone else knows a good way to explain this please chime in.

Suffice it to say that you are correct in assuming that weapon damage does not affect magicka-based skills. Basically any weapon, be it dual wield, two handed swords/axes or staves and their stat values add to both the weapon damage and spell damage stat pools. You can see this by stripping your character naked and equipping a staff or two handed sword and checking both your spell damage and weapon damage after equipping each one. Both stats will increase by the same amount when you equip each weapon. Dual wield compared to one sword and a shield produces a higher stat gain for some reason I'm not certain about. I think equipping the two swords have some inherent multiplier bonus of some kind that isn't obvious. Again, anyone who knows for sure why this works the way it does will be better suited for giving a definitive answer here.

On a somewhat related note. Sword and shield on non-tank builds is most prevalent in pvp. Ie. Dps or healer toons running sword and board on their back bar for more survivability when they need it. It would also be fine for pve questing and overland content like delves, normal dungeons and public dungeons, etc. I've yet to see a good damage dealer build running sword and shield in any endgame pve content and being effective though. Sometimes you see a healer running an off tank setup meaning they heal primarily but swap bars to be able to tank some incoming damage from time to time when the situation calls for a second tank. People who can do this effectively are pretty rare in my experience though.

I've rambled a lot here but hopefully something I've said will help you out. Best of luck in your character building.