I don't think it fits the sport category. It's a game. One that can be played using money.
All sports and games can be used as a format for gambling. Make sure to do so responsibly.
Poker is a game that commonly uses a monetary system to keep track of position in game. But it is not necessarily necessary.
slots machines and
roulette are straight up gambling games. No real tactics, just bet. Like a lottery. Bet, Win, Lose.
Gambling on the dogs or horses, 1st past the post. Though you have more information on the animals. They do the sport we play the statistics game on that. Bet, Win, Lose.
Standard team sports bets tend to have more bias as you tend not to bet against home. Bet, Win Lose.
Poker (live especially) is a game closer to chess. Being much more involved in the games development as it progresses. Deciding to make a move or not.
Games are generally mentally exhausting.
Sports are generally physically exhausting.
Pub games such as snooker, pool, darts are a bit of both. Still i rate them more in the game category, though they seem to be getting more recognition as sports these days.
Even stocks are played like games these days.
Gambling doesn't just apply to sports and games. And it isn't always monetary.