Duration comes into play here. Is blackjack a single long game or is each hand a "game?"

The house isn't a player. They offer the game for players to play. That is a fundamentally different role.

Saying that the house is playing the game against you is equivalent to saying that Nintendo is playing Super Mario Bros against you.

There is an entire category of casino games where the house is a player. These are called house-banked games and they include most of what you imagine when you think of a casino floor: blackjack, roulette, craps, slots, etc.

But even in player-banked games like poker, the house is an agent participating in the game through the dealer. They can still exercise their right to free association and not play with you.

A corrected analogy would be Nintendo banning you from Mario Kart Online for suspected cheating. Yes, Nintendo isn't playing, but they still have that authority as the facilitator.