Here the question of hypercorrection becomes difficult. Although axiom, rhinoceros, and octopus are originally Greek, the OED etymologies indicate that all three come to us through Late Latin or through the scientific Latin of Linnaeus and his time. Penis, of course, is Latin, not from Greek.
Axiom comes from Gk. axioma, pl. axiomata. Borrowed into Latin it retains the form axioma with the gen. sg. axiomatis. It's a neuter noun in Latin as in Gk., so its nom. plural is axiomata. Forms of the word occur 3.4 million times in Latin texts (from Perseus), to me a surprisingly high number. Axia would be an analogical formation based on criterion/criteria etc. OED does not list axia as a reported plural form, suggesting that it is recent and may in fact be a hypercorrection.
Penis, pl. penes (UK), penises (US), (irreg.) peni, is originally Latin and enters English in the late 16th c. as a scientific term. It is cognate with Gk. peos but is not borrowed from Gk. Peni may also be a hypercorrection based on the fact that unstressed -is and unstressed -us sound the same, so the -is ending gets treated like a Latinate masc. nom. sg. suffix whose plural would be -i.
Rhinoceros is derived from Gk. rhino "nose" and keras "horn." The plural of keras is kerata, but the compound form changes the short alpha of the singular to a long omega, with the plural rhinocerotes, also with omega. Latin borrows this, and in Med. Lat. the plural rhinocerota is found, changing the gender from Gk. neuter to Latin masc./fem. While the OED doesn't give plural forms for rhinoceros, the -i plural arises in the same way as for penis. Gk. -os corresponds to Latin -us as the masc. nom. sg. whose plural is -i. The Gk. plural for the -os suffix would be -oi, which English uses only in the borrowed phrase hoi polloi "the many." Otherwise English uses -i for the plural of originally masc. Gk. nouns in -os.
Octopus comes from Gk. oktopous, "eight-footed." Its plural follows the Gk. as octopodes, although that's now found only in formal scientific usage. Octopi is much more common and in this case the explanation is more straight-forward. Octopus also comes to us from Linnaeus and therefore as a Latinate form, separated from its Gk. roots. The -us suffix is easily reanalyzed as a Latin form whose plural is -i. But here we get into the question of where legitimate reanalysis shades into hypercorrection. Hypercorrection normally involves social insecurity. I'm not sure that's the case here.
One of my favorites is "syllabus," where the picture is rather messier. I've been in more than one faculty meeting where some colleague insisted on the plural "syllabi" rather than "syllabuses." OED does give syllabi as the first choice for the plural, but I'll quote the OED directly on this, changing only the Gk. letters that I can't produce in email:
"[mod.L. syllabus, usually referred to an alleged Gr. syllabos. Syllabus appears to be founded on a corrupt reading syllabos in some early printed editions--the Medicean MS. has sillabos--of Cicero Epp. ad Atticum IV. iv, where the reading indicated as correct by comparison with the MS. readings in IV. v. and viii. is sittybas or Gr. sittybas, acc. pl. of sittyba, parchment label or title-slip on a book. (Cf. Tyrrell and Purser Correspondence of Cicero nos. 107, 108, 112, Comm. and Adnot. Crit.) Syllabos was gręcized by later editors as syllabous, from which a spurious syllabos deduced and treated as a derivative of syllambanein to put together, collect (cf. SYLLABLE)."
So syllabus is a late development and is not etymologically Gk., although it's based on Gk. roots.
Etymology gets to be great fun. Sometime take the time to work through the lengthy OED etymology of "church."
Herb
I just came across the hypercorrect Greek and Latin plurals axia,
peni, rhinoceri, and octopi in Pinker's _Words and Rules_.
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