Good question. I think the existence of an 'identity' and the existence of something like a 'soul' are different, in that identities are necessarily subjective. In the article, I compared gender identity to identifying as an Australian or a Canadian, which of course exist as subjective concepts too. What this means is, you cannot objectively prove that an 'Australian identity' exists, except by accepting the subjective identification of Australians, for example. Therefore, for the category of identities, I think the level of 'objective' proof needed would just be observable identification by a significant number of people.

Comparing 'gender identity' to 'souls' is therefore like comparing apples and oranges, but I guess we can compare it to 'Christian identity'. We can accept that 'Christians' exist despite being unable to prove the existence of souls or God, because 'Christian' is an identity label, and its validity is established simply by a group of people consistently and observably adopting that identity. The 'Christian' identity hence objectively exists, even though their religious beliefs still can't be proven to be objectively true.

In fact, gender identity has an additional ‘proof’ of being valid: 1) gender dysphoria is a medical condition proven by the standards of clinical medicine; 2) those with gender dysphoria do not have any psychotic delusions (it is part of the diagnostic criteria); 3) the vast majority of those with gender dysphoria have a strong gender identity. Therefore, gender identity is part of a condition that is proven to exist based on the long standing standards of clinical medicine.