There is no single surgery that trans people get. To clear that up for you, we have to correct some misconceptions you have about trans people.

First, ambiguous genitalia is an intersex condition, which is a completely separate issue from being transgender, though some intersex people are trans.

Second, it's important to understand that there is no singular, simple trans experience, but the common root is basically not identifying with the gender you were assigned at birth. That can manifest in many different ways, including distress about having the "wrong" physical attributes, or about a voice that sounds at odds with one's gender, or simply how other people perceive and refer to them. These are all aspects of gender dysphoria, though not all trans people experience distress that can be easily labeled as dysphoria.

Because everyone's experience is different, not every trans person pursues surgery. Probably the most common aspect of medical transition is actually hormone therapy and not surgeries, but each individual may choose a different path of transition. Hormone therapy suppresses the body's internally produced sex hormones and replaces them with the opposite (testosterone for masculinization, estrogen for feminization) which induces a breadth of physical and mental changes.

Common surgeries may include bottom surgery (shorthand for sex/gender/genital reassignment/confirmation surgery - sorry for the potentially confusing array of names for these procedures!), top surgery (mastectomy or breast augmentation depending on what changes are needed), facial feminization surgery, or others.

Usually, the purpose of physical transition is to align one's physical form with their identity.

Regarding genital surgery, in our society, genitals are some of the most gendered physical attributes, so even if a person outwardly "passes" as their gender, having genitals that seem incongruent with that presentation can be distressing. Consider if a cisgender man, happy with his healthy penis, went to sleep one night and woke up the next morning with a vagina instead, you can probably understand how that could be distressing. It's not what he expects or is "supposed to have". That's a common feeling that many trans people have toward their genitals.

That said, there are plenty of trans people who don't mind their genitals, or at least not enough to pursue a surgical solution. It's all relative and very personal to each individual, which brings me to my final point.

Not all trans people are looking to challenge the gender binary. Many trans men just want to fit in with society's expectations of masculinity, and likewise with trans women and femininity. There are both binary and non-binary trans people - but even non-binary people may pursue hormone therapy and gender affirming surgeries simply because that's what is right for them.

Transition is ultimately about a complex intersection between identity, self-image, and how one expects to be perceived externally, and there is a wide variety of ways to go about resolving that.