No, with the incorrect one.

actual gender

Trans people are born trans and are biologically the get they identify as.

Here are a number of studies that show the biological reality of transgender people. It’s important to note that although some of these studies suggest causation (ie, HOW a person is born transgender) we can’t know if there’s more than one cause, and therefore using any one test to see if somebody is “biologically trans” is impossible.

Sex vs Gender and the role of the SRY gene specifically (as opposed to the Y chromosome generally) in the development of each, as well as the possible role of AR (androgen receptor) gene https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6505576/

A correlation is found between AR allele repeat length and a person being transgender https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3402034/

The brains of trans people more closely resemble those of their chosen gender identity than their natal sex https://www.ese-hormones.org/media/1506/transgender-brains-are-more-like-their-desired-gender-from-an-early-age.pdf

A trans person's brain more closely resembles their chosen gender as demonstrated by their brain's response to olfactory stimulation with androstadienone https://www.frontiersin.org/articles/10.3389/fendo.2014.00060/full

Estrogen signaling pathways identified which tell the brain to develop as one gender or the other, as well as mechanisms by which physical development can happen in one way while brain development can happen in the other https://www.nature.com/articles/s41598-019-53500-y

Cornell University summarizes 72 studies regarding the effect of gender transition on transgender well-being (with links to all 72 studies) https://whatweknow.inequality.cornell.edu/topics/lgbt-equality/what-does-the-scholarly-research-say-about-the-well-being-of-transgender-people/

So no, you've got it backwards: they're being forced to turn in their id with their actual gender in exchange for one with the incorrect gender marker.