"Sex-based rights" is not a slogan, it's a descriptive term, although a perhaps more appropriate term for most of the individual issues would be "sex-based segregation or sex-based distinction.
I think that sports is a pretty straight forward place to illustrate this. But it must be taken as given that sex and gender are seperate, but related concepts.
Does the segregation is sport exist because of sex (biology) or gender (psychology/sociology)? Is it due to the difference in physical attributes and capabilities between two populations or is it due to social concerns: "Boys and girls shouldn't play together!" Arguably, there is a combination of both in play.
I think that there is no longer justification for social segregation in sports. The idea that girls and boys (men and women) shouldn't play together is obsolete and should be discarded. The only rational reason is the differences in physical capability between males as a population and females as a population. So I think the segregation is primarily sex-based.
So the "right" to sports league segregation could be termed as a "sex-based right."
I will note that some people argue that trans advocates, including some athletes, argue that there is variation within the populations and therefore trans-women don't really get an advantage. This indicates a lack of understanding of populations and statistics. This argument does not support the idea that a particular trans-woman (or trans-women in general) belongs in the "female" population. Rather it is really more of an argument that there are not two populations at all. Neither side wants that.
If there is a single population, there is no rationale for segregated leagues. But we know from the statistics that a single set of leagues will be dominated by males, with relatively few females.
Now, an argument could be made on an individual case basis that treatments associated with transition might sufficiently remove the advantage such that a particular person might move from one population to the other. The problem is that this is very hard to test for. And what kind of performance change should be required anyway? A percentage somehow measured in good faith both before and after transition? Performance at a level where the athlete is unlikely to win? A certain amount of time? Thus far, no one seems to have come up with a very good formula.
Sometimes athletes have to make choices between sports and healthcare. For example, many of the drugs that are outlawed in sports as PEDs have therapeutic benefits for certain conditions. At the time my daughter played softball, some medications for ADHD were not allowed (adderall).
Another area where "sex-based rights" comes up is something Emily's Cat often mentions: scholarships and set asides for women. These are things that were designed to help women overcome discrimination and sexism and get them into certain fields or guarantee a certain level of representation on certain committees, boards, etc. Note that these are pretty much socially derived.
The argument here is that these programs were created to battle sex discrimination and that giving them to a male (trans-women are still male by sex) the objective is circumvented. I'm a little fuzzier on this, because I think the objective of these programs should be that at some point they will have accomplished their missions and then go away. Also, many of these things, such as scholarships, are offered by private organizations who can make and modify their rules as they see fit.
But the idea is that if, for instance a law was passed in 1975 that at least 3 of ten members of the governing board of a public univesity must be women, the intention was to give women guaranteed representation to avoid or overcome sex discrimination. A transgender woman filling this role would not meet this requirement because they are not of the sex in question and have not faced that discrimination. Obviously, it's not that they have not faced discrimination, but that they have faced a different sort of discrimination and therefore bring representation of different concerns to the table. So if one of the three "woman" board menbers is a trans woman, the female representation is only 20%, not the intended 30%.
And then the right to segregated spaces. Are the segregated spaces segregated by sex or by gender? Depending on why you think we have segregated spaces, you either think these are sex-based or gender-based rights. Or you think that the whole segregation thing is outdated and we should be moving away from it entirely.
Does this give you an idea of what is meant by sex-based rights?
Note that thinking that any of these areas are sex-based and not gender-based does not imply any animus, dislike, or disrespect for trans people or the validity of their identity, just that in certain cases, it is about sex, not gender.
The fact that people who are transphobic will hold these positions does not make the positions transphobic. Nor does it follow that one is transphobic for holding these positions. Just as the fact that voyeuristic men might support self-id as it gives them easier access to unclothed women does not make the idea of self-id misogynistic. (I don't think most trans people who support self-id do so out of a desire to hurt women.)
This is getting long. I didn't intend to write an essay.
