"Cancel Culture" is not a real thing

No, it was not transphobic.

Your claim "the myth on society that says a man can be a woman" is absolutely transphobic rhetoric, yes.

When people are overreacting and getting offended to something that is factual, that sounds a lot like bigotry

Not only is that not what bigotry means, but what you had previously said was not necessarily factual either, and even you yourself point out that being trans is not a biological characteristic, so implying that your claim was "factual" is just incorrect.

Again, it's not the same if someone is being bashed for making a valid or harmless point.

And again, "a valid or harmless point" is subjective; for example, I would obviously consider the claim I had called you out on (the transphobic one) to be neither valid nor harmless, as it was blatantly transphobic rhetoric.

Being trans, for example, is based on identity and feelings, not facts.

Why do you assume those can't be both? If someone is happy, and they tell you they are happy, then that is a factual statement, just as if someone is trans, and they tell you they are trans, that is also a factual statement. Just because something relies on identity or feelings doesn't mean it can't also be factual.

Either way, humans need to seriously stop messing with nature.

Given the context, I assume you mean to imply that being trans is "messing with nature" and that people need to stop [being trans], and if my assumption is accurate, then this is again an incredibly transphobic statement.

I don't base things on your opinion of what's true or not.

Except in instances of identity or feelings, right? Because those things are fundamentally subjective, so you cannot say or prove what someone else's identity or feelings are to begin with. If someone tells you they are happy, then you have no basis to insist that is untrue, just as if someone tells you they identify as a guy, then you have no basis to insist that is untrue. In cases of identity and feelings, there is no objective basis of truth other than that person's own "opinion", so who exactly are you to insist you know "the truth" otherwise? After all, if you recognize that identity is, as you explained, "a social phenomenon, not a biological one", then what is your issue with? You understand that gender is social, not biological, right?

But you can't ignore that there is real shoving of controversial ideals that don't belong at schools (especially amongst younger children).

Oh of course there is, like Florida's recent education laws that promote historical revisionism when teaching about US history and politics, or private religious schools that can deny science and history in favor of teaching religion and creationism (which I attended as a child), etc. Those kinds of things are definitely propagandized indoctrination, and should be abolished.