"This card cannot be Special Summoned except by Synchro Summon." is not treated as an effect.
The effect targets the 1 or 2 cards to be destroyed.
After the effect of "Trident Dragion" resolves and it gains the ability to attack multiple times, if "Skill Drain" is activated, then the effect has already resolved so "Trident Dragion" can still attack multiple times.
The text "in addition to its normal attack, this card can attack a number of times equal to the number of cards destroyed by this effect." of "Trident Dragion" is part of the effect which activates when "Trident Dragion" is Synchro Summoned and destroys a card on your field. This effect is a Trigger Effect.
When the effect of "Trident Dragion" activates and targets two cards, if an effect is chained so that one of the two cards is not on the field at resolution, then the remaining card on the field is still destroyed. Furthermore, "Trident Dragion" can only attack one additional time.