This issue is nothing new, it's been around since we as a species have been around
What is the evidence of this?
Plenty of examples in ancient Greek and Roman texts, most classics students would be aware of this. I have also heard of examples of this in ancient Egypt and the Americas, though I'm not familiar with those areas as much. Quick google search shows that it's been recorded across a whole range of cultures dating back to ancient times. So it's definitely something that's been around for a long time.
I studied classical history for two years, I learned all about the primary role of homosexuality in Roman and Greek societies, yet I never saw one single thing about people believing they had been born the wrong gender, and that their brain and thoughts were incompatible with their bodies and sexual organs they had. In those societies homosexuality (or more accurately, paedophilia) for males was usually a rite of passage. As a young man it was expected that you would be mentored by an older man, who was heterosexual and had a wife, and you would be expected to sexually service him.
Despite that, when you came of age, you were expected to be heterosexual, and to marry and have children.
The problems a system like that causes is a whole new conversation, and is irrelevant here, as you've pointed out yourself - we're not talking about homosexuality, we're talking about gender dysphoria.
I don't see any history anywhere of gender dysphoria being a widespread thing, until the late 20th century. What did you google exactly?