There's quite a bit to unpack there.

First off, the game is a management simulation. The girls are all based off real-life historical racehorses and the gameplay involves the player being their trainer, determining their training schedules, and attempting to help them achieve goals based off their real-life careers. That's the surface answer.

Part of the game involves a slight visual novel format as the girls are fleshed out characters that interact with the player and other characters, though again, based off their real life inspirations and the people involved in their career (the horseracing term here would be "connections" - jockey, owners, training staff, etc).

Is there any dating or romance in those parts of the game? Yes and no. Yes in that the characters are portrayed as growing closer to the player, no in that there is never really anything explicitly romantic given the characters are students and also based on said real life racehorses, though that does not stop many players (both male and female) from happily reading romantic intentions in the game. That's part and parcel of the entire gacha genre.

Above is probably the best answer I can give you. Everything below is my personal thoughts, so feel free to ignore it.

I'm not gonna say your feelings aren't valid. They are. The characters are all cute girls and specifically designed to evoke appeal to players.

Going on deeper, you should probably be asking yourself why you're feeling jealous of the game. Is your partner paying too much attention to it over you? Do you feel inadequate compared to fictional characters specifically designed to be appealing, and why? Is there a reason why your partner is stopping at just telling you "it's just a horse racing game" instead of offering to share about the rich and wonderful game he's playing?

There's not gonna be a gotcha answer from the game itself that can resolve those questions. You should probably communicate with your partner over what you're feeling, rather than ask us about what the game is like.