An intersex person is typically assigned a gender at birth, but so is everyone else. Being intersex just means you aren't biologically male or female (though I think this might also include people who have sex chromosomes that develop as though they were the other binary sex, but I'm not an expert). Most intersex people don't typically know they are intersex, and thus they would count as cisgender so long as they identify as the gender they were assigned at birth and transgender if they do not. Thus, if someone had, say XY chromosomes, but was assigned female at birth, they would probably be cis if they identified as female.

However, trans can be a bit of a self-identifying label, and thus someone in that situation might just as well consider themselves trans. There's a lot of different definitions for trans. Many non-binary people would consider themselves trans since they don't identify as their assigned gender at birth.

Long story short, gender is complicated. Sex doesn't change (put a couple asterisks here), but gender is super flexible (also asterisks here.)

Well intersex could be either, both or none. Some are born with something that looks like a vagina and has balls inside of them so they think they are biologically a woman until they try to have kids and it isn't possible or are together with someone who knows what a vagina "should" look like or it looks kind of like a penis but everyone is a bit unsure. It is a big grey zone, I don't know if you can even call them biologically anything, if biologically means having one type of genitals. I am not sure if bilogical means anything either way (for intersex). It depends what you want to actually know, if you want kids then no, no biologically kids. But that could be true for someone who is what you would call biologically man or woman any ways.

So yeah one day they may discover they can't have kids or that they also have balls inside of them, or they have no balls or they have a mix of both or none. Before (and still in some countries) did the parents pick a gender and then a lot of surgery happened to make it look like a typical penis or vagina.

But it is an interesting topic, what is even sex? What do you wish to actually know? Because if it is about kids then sex doesn't matter instead it would be better to just ask "can you produce x" ( depending on what you produce). Not everyone who is "biologically" can produce what you need anyways. I know many who needed to get a sperm or egg donator (or both) to have kids.