You're right that in French, je joue is equivalent to both "I play" and "I am playing."
But when you want to emphasize the fact that you're playing right now, you can use the expression être en train de jouer.
Jouer is in the infinitive because that's how French works: when you have two verbs (in this case, être and jouer), the first is conjugated and the second remains in the infinitive.
You can also use être en train de with other verbs: être en train de manger (to be eating), être en train d'étudier (to be studying), etc.
I hope this helps, bonne continuation !
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