No, it isn't true.
Our society is based upon individualism. The right of each individual to live as they choose, where legal restrictions are based upon demonstrable social need rather than dogma.
There is no demonstrable necessity for sex based restrictions, legal or social, on clothing for example in everyday life. What the justification of these restrictions boils down to is an insistance that each of us presents ourselves as being appropriately, sexually available.
I read about police officers in the US, for example, making issue with drivers who appear to be female but have M on their licience. The driver has confirmed their identity. If further proof of identity is necessary, then that might be provided. The insistance of the police officer is tantamount to sexual molestation.
If the driver presented with red hair but was listed as being blond, would the same problem arise?
But my outward gender is incompatable with my feelings. It's more than appearance. It smells, it hurts, it affects my behaviour and judgement, (less as I've grown older of course), it makes it difficult for me to relate to others.
That is not a social construct, it is biological.