I would usually say something along the lines of 「マリオを遊ぶ」but I've been informed that 「マリオする」is a more natural say to say this phrase. Doesn't make sense to me, so I was hoping someone could explain
You can say:
遊ぶ is relatively wordy and has a nuance of "have fun", so hardcore/serious gamers may tend to avoid 遊ぶ and simply say する/やる.
A specific title like マリオ does not work as a suru-verb, so you can say マリオする only in highly informal conversations where particles are often omitted (and in such informal settings, you can also say マリオ遊ぶ, anyway). ゲームする is becoming tolerated recently, but it's still better to say ゲームをする in formal articles.