In my experience, what you find when speaking with intersex people about this is an interesting split based upon gender identity. Intersex people who do not identify with the binary sex they were coercively assigned at birth tend to see all intersex people as forced to live trans lives. Intersex people who do identify with the binary sex label they were given at birth instead generally see themselves as cis people, and only frame as trans gender those intersex people who gender transition or who assert a nonbinary gender identity.
Personally, as an intersex gender transitioner, I fall into the camp that does not view intersex people living in our society as cis gender, even if their gender identity matches their assigned sex. Intersex children are born neither male nor female, but are forced into a binary sex category by a contemporary social ideology that says this is mandatory. Many are then subjected to infant sex assignment surgery to try to make their bodies conform to their assigned sex. What is that other than a forced sex change? Just because a person grows up to identify with the sex they were assigned at birth does not mean they will feel surgeries they were subjected to were appropriate. Loss of potential fertility and loss of capacity for sexual sensation are prices that they may not consider worth the result of a somewhat-more-sex-conforming body--note that many people who gender transition by choice choose not to get genital surgery. Thus, I believe, framing medical interventions into the reproductive organs and genitals of intersex people as trans interventions, not "corrections," is important, as it will force doctors to give us agency over what is done to our bodies, and prevent them from removing the very sexual features we may most identify with.
The problem with my framework politically is that a majority of intersex people today do live in their assigned binary sexes, growing up as we do in a society in which that is the norm. The percentage of us who mature to gender transition or assert a nonbinary gender identity is much higher than is the case for nonintersex people, although we don't know the exact degree of the difference because doctors are emphatically not collecting data on us, their sex-assignment "failures." Still, a majority do live their lives in their assigned sexes (often completely in the closet about being born sex-variant, as doctors have urged parents to train their children to be). And most such people do not identify at all as "forced to live a transgender life." That is, they identify as cis gender.
If someone says, "I was assigned female (or male) at birth, and I identify as female (or male)," then we usually call such a person cis gender. So intersex people who understand themselves as cis gender have a very valid basis for framing themselves that way. Certainly this is the way the medical field treats the situation, in claiming to assign us to what they used to call our "true" sex, and now call our "best" sex. Doctors view themselves not as imposing sex changes upon unconsenting infants, but as revealing our "real" binary (cis) sex.
I feel that understanding all intersex people who have been assigned a binary sex (which, in the US today, is all of us) as trans is useful, because it gives us a way to oppose unconsented-to infant genital surgeries. I view those intersex people who are happy in their assigned sexes as no different from people who are not intersex, but gender transition by choice and are happy as a result.
At the same time, I don't feel I have a right to tell an intersex person who identifies as cis gender that they can't do that. After all, as trans gender advocates note, every person is coercively assigned to a binary sex at birth. A person who grows up to identify as genderqueer, or with the binary sex they were not assigned, is forced to struggle with medical and legal and social forces to have their identity recognized, whether sex variant by birth or born with a body considered normative. So, viewing all cis people as coercively assigned to the sex with which they identify makes calling intersex people who identify with the sex they were assigned "cis gender" reasonable, from a trans-affirming perspective. (Of course, many people are not trans-affirming, and transphobia can motivate rejection of being labeled trans gender. But I do not believe it is either charitable or necessary to assume that an intersex person who identifies with their birth-assigned sex and rejects being labeled as trans is motivated by bigotry.)
I just feel that labeling anyone who is medically altered to change the sex characteristics of their body as trans makes the most sense, and is useful from an advocacy standpoint.
I've done some additional thinking about this topic, and would like to have people consider approaching gender identity in intersex people by acknowledging that we can never address intersex experience well through binary terminology. What we may really need to do is to introduce another term.
what I would suggest doing is adding to the terms "cis" and "trans" another term often used in scientific terminology. In chemistry, which gives us the language of cis and trans isomers, there are chemicals based upon a ring structure, called arene rings. When a chemical substitution is made in the same place on the ring, this is referred to as "ipso" substitution.
If we were to add the term "ipso gender" to trans and cis gender, we could perhaps describe intersex experience more accurately. A cis gender intersex person would be one with an intermediate gender identity, since that "matches" their birth sex. An ipso gender intersex person would identify with the binary sex they were medically assigned (the social sex substituted for their intersex birth status being the same as their identified sex). And a trans gender intersex person would be one who identifies with the binary sex other than the one they were assigned by doctors.
This terminology solution is not without its drawbacks. Usually people who are genderqueer in identity are considered to fall under the trans umbrella, but in the case of intersex people, they'd fall under the cis heading, which could prove confusing. But it's also possible that confusion would itself prove productive.
It's certainly worth considering.