Your grammar question looks deceivingly simple indeed, but I must tell you that it’s much easier asked than answered! I therefore had to give it a lot of thought before coming up with an answer.
First, let’s make the distinctions clearer between the four constructions by writing them down as complete sentences:
(1a) “We are all unique in our own way.” (1b) “We are all unique in our own ways.”
(2a) “We are all unique in our own right.” (2b) “We are all unique in our own rights.”
Offhand we can see that the first three constructions—1a, 1b, and 2a—are undoubtedly correct grammatically and semantically. Only 2b, “We are all unique in our own rights,” looks and sounds suspect, and about this I’ll have something more to say later.
The next thing we will discover is that the adverbial phrase “in our own way” is used in Sentences 1a and 1b in the literal sense of “in our own personal manner as an individual.” One grammatical consequence of this literalness is that the noun “way” in that adverbial phrase can take either its singular or plural form depending on whether the subject of the sentence is singular or plural, as in these constructions: “I am unique in my own way.” “She is unique in her own way.” “They are unique in their own ways.” “We are all unique in our own ways.” (Of course, if the speaker means to use “way” collectively for the entire group, this latter construction should use “way” in the collective singular form and knock off the adverb “all”: “We are unique in our own way.”)
Now, in sharp contrast to this literal character of “in our own way,” we will find that the adverbial phrase “in our own right” is actually an idiomatic expression in the sense of “by virtue of one’s own qualifications or properties.” Recall now that one major feature of an idiom is that its key or operative word isn’t substitutable or modifiable, so that changing the way the words of an idiom are put together or inflected alters its meaning. In the case of “in our own right,” the figurative meaning of the idiom is lost when the noun “right” is made plural; indeed, “rights” has the different sense of “the interest that one has in a piece of property,” as in “We own the mineral rights to that piece of land”). This is why the sentence “We are all unique in our own rights” is defective both grammatically and semantically.
As to your last question on what to call the word “way(s)” and “right(s)” in the sentences you presented and what their function is, I really can’t figure out how to answer that. All I can say is that “way” and “right” are nouns integral to the adverbial phrases in question, and as such have no independent grammatical function of their own.
