All of your statements apply to a pre-transition trans man (I mean a man who dresses as a woman, so maybe trans woman is the right term).
There is no biological basis for it. It is purely how he (she) feels on the inside.

We don't know that. There's no morphological difference that corresponds to trans identity - but the same is true of being gay, also. We don't know whether there's a genetic or inherited component to being gay or trans, or whether there are biological manifestations of being gay or trans that might show up in neurological or other physical brain phenomena. There are lots of psychological characteristics that we suspect might have a biological component because of their tendencies to be inherited, even though we can't directly map them onto an "anxiety gene" or an "extroversion/introversion" lobe of the brain. Indeed, psychologists and psychiatrists suspect that all of the "Big Five" personality aspects (openness, conscientiousness, extroversion, agreeableness, and neuroticism) have a heritable biological component, but they're not sure how that actually works.

It certainly manifests in a person's internal sense of self. But invariably for someone to actually be diagnosed with gender dysphoria you will find that they have sustained, longstanding thoughts and feelings that constitute that condition. They don't just wake up one day and decide that today they're going to be trans.

This I agree with. Nonetheless, the new Jewish person has to start somewhere. And that's how they can defend their choice to the court. Never mind that this only applies to basically honest people, and not criminals without qualms about lying to the court (as some trans prisoners have done).

The fact that some people might falsely claim to be trans doesn't mean that anyone accepts that someone's mere declaration of being trans (or being black or being Jewish or being gay) has to be taken at face value. If a person who has never, EVER given any indication external indication that they've had gender dysphoria or trans identity suddenly claimed to be trans, we can certainly take that with a grain of salt until they've actually started "walking the walk" and living their claimed identity to see if its real or pretextual. Again, it takes more than saying "I identify as X" for you to be X - whether X is "gay" or "Jewish" or "trans."

But trans advocates are basically claiming that ONLY the intent matters, biology does not. I can't imagine why it has to be true only for trans people, and not others.

Again, we don't know whether biology is involved. Because being trans shares many of the same characteristics of being gay. It manifests as a psychological orientation, not a group affinity or identity like being a member of a race, nationality, or ethnicity. There are differences between race, nationality, ethnicity, gender, sexual orientation, and other identity markers - they're not all the same, and it's not surprising that one's internal sense of self might have different importance in some areas and not others.