Is casting a commander from the command zone considered casting a spell?
Yes, they do have to pay extra mana.


  903.8. A player may cast a commander they own from the command zone. A commander cast from the command zone costs an additional {2} for each previous time the player casting it has cast it from the command zone that game. This additional cost is informally known as the “commander tax.”


(emphasis mine)

It's a creature spell, just not cast from their controller's hand as normal.


  701.4a To cast a spell is to take it from the zone it's in (usually the hand), put it on the stack, and pay its costs, so that it will eventually resolve and have its effect. A player may cast a spell if he or she has priority. See rule 601, "Casting Spells."





  701.4b To cast a card is to cast it as a spell.