The study uses gender identity as the innate sense of being male or female. It says that most people are able to identify their own gender, which is consistent with their sex, by the age of two. It says a small percentage of people (age unspecified) report significant clinical distress because their sex at birth does not reflect their gender identity; in extreme cases people will be given a diagnosis of gender dysphoria and may undergo treatment to better align their anatomy and physiology with their gender identity.
So, no, they are not saying gender dysphoria is apparent at age two, just that most children know they are male or female (and accept themselves as such) by age two.
The authors appear to accept the concept of gender identity at face value, as something innate.
Gender dysphoria is the diagnosis given to extreme cases where gender identity does not correspond to physical sex (in this study). It says early research suggested it was a psychological condition, or caused by trauma but more recent research shows their may be a biological basis with endocrinological, neurological and genetic factors. There are then references to studies which showed a higher level of gender dysphoria in women exposed to androgens in utero, for example. It says neuroimaging studies of brain scans show areas in transwomens brains closer to female than male brain scans in control groups.
The hypothesis of this study is to do with variants in sex hormone signalling which lead to undermasculinisation in men and/or feminisation of the brain.
I need to go right now, so will come back to your question about sex hormone signalling, as I need to read it more closely.
Note: I am just conveying the content of the paper, not my own views.