Thanks! Sorry if it wasn't too clear, Y chromosomes usually have the SRY gene but sometimes they don't (and there are other genes involved too, but SRY seems to be the important one that sets things in motion)
And yes basically the doctor looks at the genitalia and that's it. I think nowadays some things are normally tested (like XXY or XXX) but I'm not an expert, I don't know which genetic or chromosomal tests are common nowadays.