A right is something you are specifically owed by virtue of an obligation. The obligations differ, and as the obligations differ, the rights differ, so that, for instance, some rights are the correlatives of obligations to do something to or for you, others the correlatives of obligations for others not to do something to you or for you, others the correlatives of obligations not to prohibit you, etc.* Since obligations don't float free of everything else, but have to have some reason for why they are obligations (which we might call the principle of sufficient reason for obligations, the PSR-O), any right has to be due to the person having it being really classifiable (not necessarily being in fact classified) in a certain way. Thus, for instance, human rights are due to being human, civil rights are due to being a citizen, contractual rights are due to being a party to a contract, etc.
Human rights are in many ways the most interesting, because rights arising due to being human have a sort of priority over most other rights. To generate an obligation, though, being human has to impose some kind of requirement on other human beings. It seems that there are only two ways this can be done, either because it is required for the full humanity of the human person qua human, and thus is shared among all human persons, or because it is required for the full humanity of the human race qua human, and thus, while not necessarily directly shared, is essential for the common good of the whole. Thus, for instance, rights of conscience are examples of the former; many rights protecting children or the elderly, like the right of a child to the protection of its parent or guardian, or the right of the elderly to care, are examples of the latter.
Unfortunately, because rights have a rhetorical as well as jural function, people have a tendency to muddle these together as it suits them; this is common with a lot of rights-sloganeering, that is, cases where, instead of actually making a case, people instead use an advertising slogan whose actual relevance is not developed properly. A recent one that has been common is the slogan 'Trans Rights are Human Rights'. Like many of the slogans that have been used in recent times in place of real ethical positions, this slogan admits of a possible and obviously true sense, and also of a more apparent but controversial, false, or incoherent sense. It could, on the one hand, just be a misleading way to say, "People classified as trans have the same human rights as everyone else," in which case it is certainly true, but doesn't actually get anyone very far, because there is nothing specifically trans-related about any of this; it applies to any group of human beings, however arbitrary. It is also not the most natural way to read it. The most natural way to read it is to take it to mean that rights specifically concerned with gender transitioning are human rights. And then we run into the problem that human rights are not themselves concerned with any specific group of human beings at all; they are common to humanity, either in that everyone has exactly that right, or the community of everyone depends on that right existing.
If X is a human right, in order to know how to recognize it properly as such, we need to know the reason for it so that we can use the appropriate classification (by PSR-O), and tailor our actions accordingly. But a human right has to be something that is either specifically due to some feature of human nature as such, like reason or needs for physical survival and thus shared by all human beings, or derives from some feature required for the humanity of the human race itself, like the need to have societies or to reproduce in a way that makes it possible to live as human beings. Those are the only two things we could actually mean by the 'human' part. It is entirely contrary to the point of the label to take 'human' to mean less than what belongs to all human beings. Now rights specifically concerned with gender transitioning, like the right to transition, are not based on something shared by all human individuals; it is not like (say) the right to speak your mind honestly and in a reasonable way, which is based on something required for literally every human being to live a human life. So it's by definition not an example of the first kind.
Thus the only kind of human right that could be in view are rights that are required for the existence, as human, of the human race. This foundation does give us rights that, while the foundation is something shared by all human beings, specifically regard particular groups. But, in contrast to rights that have to do with children or relations between the sexes, it's difficult to point to anything specifically trans-related that is necessary for the human race itself. There does not seem to be any sense at all in which gender transitioning is necessary for the survival or coherence of all humanity; outside of drag, which by its very nature is based purely on cultural norms, and eunuchry, which despite occasional cultural importance plays no essential role to humanity at large (and the imposition of which has usually been considered a violation of human rights, even when done consensually), most of the things associated with it -- hormone treatment, surgery, and their effects -- are relatively recent phenomena or are obviously culture-dependent. The human race does not degrade into chaos without it.
One could perhaps argue for a synecdoche here: the actual human right is some more general right, of which trans cases are just particular cases that sometimes arise under particular conditions. But this means that we need to know what the general human right is, and we are still going to run into the same problem: it would have to be a right essential to the human race qua human, and there would have to be some reason to think that this particular case is sometimes necessitated by that right. There doesn't seem to be anything that fits the bill.
Sometimes we use 'human rights' not in a synecdoche but in a metonymy, and we take it to cover things that are not required by the human race but valuable for its well-being. In this sense, for instance, one might say that there is a human right to literacy; this is not literally a human right, but general literacy is something that greatly enriches the whole human race, being an assistance to social relations and a source and perservative of good things that all can enjoy, like literature. This is probably the best ground on which to defend the slogan, although I've yet to see anyone use it in a way that would make sense of this interpretation.
But the fundamental problem is that ethics cannot be done by slogan. Even if the slogan is right, by simply repeating it, you have not done anything just, you have not actually helped anyone act with deeper understanding; you have substituted an advertisement for the actual communication by which we help each other to be ethical. One might as well be doing ethics by football chants. What people need are reasons, not your enthusiasm for a hashtag. And in the case of human rights, what is relevant is not the label 'human right' but the reason why it is supposed to be a right grounded in what all human beings share.
* Sometimes in the Hohfeldian system of rights analysis, 'privileges' or 'liberties' are defined as A has a privilege or liberty to do X if and only if A has no duty not to do X, which attempts to identify an 'incident' of right entirely in terms of nonduty; but this is in fact illusory, because the domain here can't be unlimited. It makes no sense to say that I have a privilege or liberty to fly to Alpha Centauri, for instance. It is true that I don't have any identifiable duty not to do it, but I have no right to do it, either. It's just not on the table as a possible action. And this problem extends to things that are not strictly impossible but nonetheless just not in view in any relevant decision-making. The only reason a nonduty becomes relevant to rights is if it is a nonduty specifically designated by some sort of obligation. To be considered relevant to rights, privileges or liberties cannot be taken formlessly, but only specifically, and something must be constructing the relevant kind of nonduty. Not only is this just common sense, it follows directly from the standard Hohfeldian view of the incidents combining to form complex rights, which means that each incident has to be picking out something definite rather than indefinite.