As someone who had accommodations in College (and the last year of high school), I never understood why people are so hesitant or against saying why. For me, with the issue of my traumatic brain injury and then subsequent brain surgeries, I think talking about it and explaining it to my prof helped both of us figure out how to best meet my accommodations. I had professors who came up with other ideas/ways to help that I never thought of that worked way better, especially in a subject like math.

As you said, talking about this stuff can be really uncomfortable, but in my experience, to get the best help, you have to talk about it. Most of my profs never had a clue what it was like, but I did have a couple who had family or friends who had dealt with a somewhat similar situation, and it immensely benefited me. Life is uncomfortable, and it's not like you need a play-by-play or an in-depth explanation, but some basics go a hell of a long way!

*edit
Yes of course someone could take the information and abuse it, but at the same time if a prof was going to do this or it crossed their mind, that is a whole other issue about that person honestly not needing to be an educator