In the commercial it showed a group of kids playing as their Amiibos, so I bought an Amiibo resembling a DLC character, however I found out that I couldn't play as the Amiibo. I can play with/against them, but not as them. Isn't there a way to play as my Amiibo?
Unfortunately, no. They are known as figure players, and they are controlled by an AI, which is modified as it learns. It battles without any input from you, though it learns strategies as it is fought. Amiibo are allies or opponents, not players.
Sadly, some of the features that were shown were never implemented, such as being able to tap them in mid-match and using them in online play. This is one of the features that didn't make it. Though I think Nintendo may have done that for reasons, like tapping them in mid-match could give you an edge that would unbalance the game, and all the boosts from feeding them equipment could make you overpowered if you could control them. I figured this out all on my own, both from the help section titled "What's an Amiibo?" and my experiences with my Pikachu Amiibo.
If that answer was too long and complicated the short version is: No, you can't. They are computer controlled characters that learn from experience, and they were going to be able to be controlled, but that idea was scrapped, along with the ability to tap them in mid-match and being able to use them in online play.