In a recently closed post Is equality as a political goal meaningless?, the OP posed this question [as edited by me]: Is the concept of political equality, itself, meaningless, without referencing the specific domain/type of equality one is addressing. This, different question, is here under consideration.
The answer to the question:
“Is the concept of political equality, itself, meaningless, without referencing the specific domain/type of equality one is addressing.”
is yes, if not meaningless, the term is so ambiguous as to be functionally useful only to political ideologues deploying the ambiguous concept as a premise in support of a desired end. (Or, for those who do not “believe in” semantic “meaning,” without further contextualizing the term it is impossible to tell when it is being used in the appropriate “language game,” amongst its proper semantic peers, etc.)
As Harry Frankfurt pointed out long ago, whenever a striving for equality is considered to be morally important, it is because doing so will promote some other value or desiderata. Which, presumably, leads us to the “types of equality” referenced in your question: for instance, equality of income, equality of [shares of] resources, equality of welfare, equality of opportunity, equality of respect, equality of consideration (political, legal, etc.), equality of concern, equality of process, etc.
His claim is that it is the promotion of these values themselves that is morally desirable, rather than that equality is itself morally desirable in one or another of these domains. This fact becomes immediately apparent if one considers, for instance, whether the promotion of equality of ill health, pain, poverty, psychopathology, etc., amongst human beings is morally desirable. As has been pointed out, income inequality might be eliminated decisively by arranging that all incomes be equally below the poverty level, but making everyone equally poor is hardly a desirable outcome. This is why the bare goal of equality in any of these domains is problematic.
I am reluctant to go as far as Frankfurt in discrediting the notion of “equality” in toto, because there is arguably some instrumental value in “equality” as a conceptual tool to achieve one or another of the desired ends. But using it instrumentally in this way is fraught with risk. For instance, as Ted Wrigley points out in his answer, it is risky to willy nilly advocate bald equality of outcomes in all domains.
Characterization of the risk can be introduced here by citing Hassan Jolany’s answer to the original [now closed] OP that"
“The sameness of A with itself [A=A] is not equivalent to the equality of A with something distinct from A. Thus A could be equal to something distinct from A by virtue of a common characteristic or a shared perspective.”
So the questions becomes which contextually salient characteristic of A can be said to be “equal” to which characteristic of B. And why/how so? This brings into view the difficulty known as the fallacy of false equivalence, commonly known as [drawing conclusions by] “comparing apples and oranges,” defined as:
An argument or claim in which two completely opposing arguments appear to be logically equivalent when in fact they are not. The confusion is often due to one shared characteristic between two or more items of comparison in the argument that is way off in the order of magnitude, oversimplified, or just that important additional factors have been ignored.
It has the logical form:
Thing x and thing y both share characteristic A.
Therefore, things x and y are equal.
For example, the claim that a knife and dynamite are both tools that can be used as weapons, so they're pretty much the same thing, and therefore if we allow people to buy knives at the store, then we should also allow them to also buy dynamite at the store.
Or consider the second paragraph of @Ted Wrigley’s answer to the original l OP here Is equality as a political goal meaningless?, where the supposed disparate "addressing" of two events, identified the relevant "characteristic" as "protest" and argued that:
"Whatever one might think of the intention and goals of these separate protests, the processes by which they were addressed were unequal."
He then implies that the two protests were addressed differently because of the identity of the protestors and "goals of the protests" differed, and that
“it is that inequality of process that leads to perceptual or actual injustice.”(See our discussion in the comments section of the closed OP).
What he is saying is that because both phenomena have the characteristic of being a “protest” (i.e., “a tool”, in earlier example), how they are “addressed” (i.e whether one should be able to “buy the thing at a store”) should equal.
Whereas, if one actually takes the time to review all of the available data/facts it is at the very least arguable/tenable, that the two [groups/sets of] "protestors" were treated differently because while both could be reasonably be characterized as “protesters,” [both objects are are tools] it is in fact another characteristic, i.e. how these protestors, qua protesters, conducted themselves, how the “protests” were manifested, that is salient to the question of how each was/should be/can be addressed, and whether or not any apparent disparity in addressing the protestors represented inequality and hence injustice.
As noted in the definition cited above:
“The confusion is often due to one shared characteristic between two or more items of comparison in the argument that is way off in the order of magnitude, oversimplified, or just that important additional factors have been ignored.” [Whether accidentally, carelessly, or intentionally in order to further an interest or cause external to the notion if inquiry].
And this, the problem of false equivalence, would, inter alia, appear to be an important difficulty in, and risk of, purporting to ascribe meaning (or use) to the term "equality" (whether in politics, science or other domain) without careful contextual eludication of its deployment. And what makes the notion "particularly vulnerable to abuse" in the outcome driven world of political discourse. And likely a reason why Frankfurt was leery/distrustful of its use.
Aside: While not all of what I have argued can be attributed to him, if any of this is of interest, I encourage you to peruse Harry G. Frankfurt's insightful consideration of some of these issues and concepts in his on inequality.]