I am an intersex person, and I'd say that the main reason is due to "proof" of being intersex. Since intersex people generally fall into the categories of having ambiguous or anomalous genitalia and sexual characteristics, this seems to give them more credibility with the close-minded, whereas with trans people, the problem is often a hormonal imbalance or mental predisposition to associating with their preferred gender. I have seen people that were religious on the internet who are completely opposed to LGBT rights except in the case of intersex people, and yet they only agreed that intersex people should either remain celibate or marry the opposite gender of the one they were assigned, so I wouldn't exactly call them "supportive". I am very empathetic to the trans community as an intersex person afflicted with dysphoria but besides that, I would like to add that although intersex people tend to gain a lot more medical credibility and support than trans people in the form of understanding, I have experienced a lot of bigoted and misinformed people telling me that merely being who I was biologically was a sin and considered blasphemy and I have been invalidated by people telling me that there are only two genders and therefore implying that I and other LGBT branches with nonconforming gender identities do not exist. I have been confused for being trans by these people despite not being trans (although, as I said before, trans people have my full support) and I was never told about my condition until I found out on my own and now I have intense gender dysphoria and it has mostly contributed to my current identity crisis.

I am sorry if this is not very cohesive, my writing skills seem to have deteriorated.