You're right that 'wrong body' is metaphorical language, and where as Mrs. Jorgensen does say “nature made a mistake” that is not meant to be taken literally - again, it is metaphorical, an attempt to help those who have not experienced what being trans is like to better understand their perspective and experiences. Have you heard a pregnant person ever say how they felt like a beached whale? Did you you think they meant that literally too?

More to the point, what actually happens is that gender identity develops through complex biological processes that sometimes don't align with other sex characteristics.

We can observe this in controlled experiments - genetically male fish and mice can develop completely female-typical behaviors when exposed to certain hormonal conditions during development. Often during the early development of the brain before sexual differentiation takes place. This shows how biology itself creates this variation.

Trans people (about 0.5-1% of the population) appear to experience a similar biological variation. That's why every major medical organization recognizes this as a real phenomenon requiring appropriate healthcare.