Making a TF game doesn't turn someone trans. Consuming and engaging with a fetish (and it's community) for an extended period absolutely will have an effect on someone's psyche and influence their way of thinking. To the point of drinking your own kool-aid.
It happens all the time with people these days, they're looking for somewhere to fit in, fall in with the sissy/trans crowd, and all these people and content they're consuming are telling them they're trans when really they're just confused people trying to figure shit out and where they fit in. In all likelihood they're probably not trans, but they feel a part of something and a community and keep with it because they get praise and support when they otherwise wouldn't have.
I don't want to use the word "cult" as it's obviously taken in a negative light, but it's the same type of shit.
Whether or not they're really trans (on an actual physiological level) isn't something easily debated (you can lie to yourself just as well as you can to other people), and obviously it's a hot and touchy topic for people, but it absolutely does happen. Sometimes people figure it out before it goes too far, others may figure it out after they've fucked up their bodies and either commit to it or swap back, and other times they kind of just fully engage with the rhetoric till it becomes their normal, oblivious to things.