The way you experience puberty is (to some extent) a result of the hormones in your body. Generally the hormones in your body are a result of your sex, however, it is possible to stop your body from producing certain hormones, and replace them with different hormones. In this way, one can have a puberty more similar to a different sex than one’s own.
Why do you think you can’t experience the wrong one? Also, unless you are saying there is only one sex, how could there only be one puberty?
It does involve hormones as part of the mechanism but puberty is primarily about the maturation of the reproductive system, and how this is experienced depends on one's sex.
To take female puberty as an example, this is the growth and development of the uterus, ovaries, labia and breasts to reach their mature form and function. Most importantly, the menstrual cycle begins, making pregnancy possible.
If a boy has testosterone blocked and is given estrogen instead, he doesn't experience any of this, except perhaps some breast tissue growth and redistribution of fat. His penis and testicles will not develop further and he will probably remain sterile.
However, he doesn't have a female reproductive system, so this is not more similar to female puberty. What he's experiencing is stunted male development, a pharmaceutically-induced eunuch state.
There is no option for him to go through female puberty rather than male puberty, because he lacks the type of reproductive system that would make this possible. As female puberty is not an experience available to him, it makes no sense to describe male puberty as being the "wrong puberty" for him.
Likewise for girls and the impossibility of experiencing male puberty.
Why do you think you can’t experience the wrong one? Also, unless you are saying there is only one sex, how could there only be one puberty?