The company that runs it also does other domains like .art, .blog, .design, .group, .ink, .llc, .photography, .style and .wiki.

I'd say they just picked the most generic thing that they thought would make the most money.

Why not .lesbian? .transgender?

.lgbt already exists (afilias)

Many in the LGBT community feel that the acronym does not accurately capture their identity. LGBTQIA+ is the currently acceptable acronym to use, so you can see how the use of .lgbt would be "harmful". Also, you can't have a + symbol in a TLD, which would also be considered "harmful" should it ever be up for implementation.

All in all, the use of gTLD names to reinforce identity is stupid. .com and .org mean company and organization. It's the words before those letters that define the type of company or organization you are, that's where your identity is supposed to lie.