Or „transsex“ if there's such English word? Word „translytis“ is used specifically for those who attempted transitioning man<->woman.

"transsex"/"transsexual" are English words, though they're seen as more archaic and non-inclusive.

I think you misunderstood what they were saying. They were trying to say that their language doesn't have two separate words to describe gender/sex, it is just one word, as they don't separate those terms.

That has nothing to do with the word "transgender" existing in their language (which I would bet exists in pretty much any widely-used language these days, most of the time as a borrowed word).

Not sure of their language, but it is the same in russian, with the single word "пол" (pronounced as "pohl") used to refer to sex/gender.