The concept of universal grammar was initially posited by 17th century philosophers, and Chomsky picked up on and technically developed that idea. In Aspects of the theory of syntax, he quotes Beattie as saying
Languages, therefore, resemble men in this respect, that, though each
has peculiarities, whereby it is distinguished from every other, yet
all have certain qualities in common. The peculiarities of individual
tongues are explained in their respective grammars and dictionaries.
Those things, that all languages have in common, or that are necessary
to every language, are treated of in a science, which some have called
Universal or Philosophical grammar.
It is thus false to claim that Chomsky's theory posits that a predisposition to learn language causes all languages are equivalent. Chomsky's particular version of the concept of UG does add a cognitive and acquisitional aspects. The aforementioned book (effectively, the launching of the UG program) discussed the distinction between particular grammar (the grammar of French, English, Russian and so on) and universal grammar, again citing the widely adopted distinction in the philosophical literature between particular grammar (or a specific language), and "general" grammar, i.e. universal grammar.
It is trivially false that all individual languages are the same. The only time that it makes any sense to say that "two languages are the same" is in a social context where one talks about "my English" and "your English", and the possibility that two individuals might have the exact same grammar therefore produce the exact same language. Typically, when people speak of "a language", they mean language in a large-scale socio-historical context whereby there are billions of speakers of "English", thus what consitutes English would be "th set of all utterances that are, will be, or have been generated by the various speakers of the language. This is a distinctly anti-Chomskian view of language, who see language as an internal, individual mental capacity.
The only sense in which all languages are "equivalent" is that there is no objective standard for evaluating languages. That is, it is meaningless to say that one language is "more efficient", "more logical", "more expressive". One could accurately say that "Vietnamese utterances tend to use more words than Kalaallisut utterances" (not that anyone has ever done such a corpus study), but that is not a value judgment.
Bee dance is also a form of communication, but English is clearly not the same as bee dance. The problem, as you can see, is the notion of "equivalent". Chomsky does not make any context-free claim that "all languages are equivalent". It is, however, probable (but not definitely provable as far as I know) that Chomsky holds that all languages can express the same unbounded range of propositions, though it's not clear how he views (communication of) emotions.
If we set aside the misconception about "equivalence" and UG, we are left with the current synopsis of the nature of the UG program as characterized in Chomsky 2007, which is determining "how little can be attributed to UG while still accounting for the variety of I-languages attained".