The game is literally listed under "Nintendo Switch Online Compatible" on their own damn website.
"Join Mario, Luigi, and pals for single-player or multiplayer fun anytime, anywhere! Take on two family-friendly, side-scrolling adventures with up to three friends"
For those 2 reasons you have every reason to believe you could play online with your girlfriend. That sucks man.
There's only 2 races: White and 'Political Agenda'
2 Genders: Male and 'Political Agenda'
2 Hairstyles for female characters: Long and 'Political Agenda'
2 Sexualities: Straight and 'Political Agenda'