Use Can I play too? when a game's already going and you want in. Your coworkers are deep into a card game at lunch: Can I play too? Spoken naturally, it's a short, friendly question that fits in one breath. Try saying it aloud now.
You're watching a game that's already started
You spot kids halfway through a kickball game. You say Can I play too? and they wave you in. That's the fit: the game started without you, and you're asking to join it now.
Your coworkers are already deep into a card game at lunch. Can I play too? tells them you know you're arriving late and you'd like to be dealt in, not take over.
The class game has been running for ten minutes and you've just sat down. Can I play too? sounds natural here because everyone else is mid-turn, and your question acknowledges that.
If they're just setting up, wait or offer help
The timing is the qualifier. Can I play too? works once they're mid-game, but if they're still setting up the board or choosing teams, it lands oddly. Say Mind if I join you? while they're arranging pieces, and hold Can I play too? until the game itself is underway.
So the check is quick: is anything already in motion? If a round has started or kids are already running bases, your phrasing is Can I play too? If people are just gathering, Are you starting a game? fits better, and you can switch to the play-too question once they begin.