Q. How could they use the Babylonian names for the months, were they not the idolatrous names of their avoda zarah? (See questions 2487-88 above)
A. As mentioned above quoting Ramban rather than remembering the Exodus, the Sages of the time decided that more appropriate was to in some way commemorate the latest deliverance from the nation they had been redeemed from. They did so by taking the very same idolatrous month names of their previous exile and turning them into Jewish names.
The name Tamuz was an avoda zarah, as it is written (Yechezkel 8: 14): And behold there the women were sitting, making the Tammuz weep. Rashi explains it was the statue of an idol, although Radak maintains it was the image of a false prophet.
The Rebbe of Rizhin zt’l (Gam Ani Odecho p. 20) added that the Sages of that time embedded kedusha into those names. Based on this principle, the Sages expounded the names of the months – e.g. Elul is an acronym for “ani ledodi vedodi li” (I am to my beloved, and my beloved is to me”), and Nisan is the month of “nissim” (miracles).
Even though the names of the months are linguistically speaking Babylonian or Persian, they were adopted by the Jews with the understanding that they were Divinely inspired names, and are also laden with Cabbalistic meanings. (See Torah Shleima 10 – 11).
The process of embedding kedusha into those names and by extension to the people returning from Babel, that was so essential to their return to Eretz Yisroel, was accomplished by Ezra, and could be the reason why we observe his yohrtzait on the ninth of Teves, when many others we don’t, such as the Avos or Aharon Hacohen.
See also questions 1650, 1654 and 2180 in regard to reciting tehilim when facing a cross in an adjoining cemetery or kidush levana on the street, since all Ontario vehicle plates have a cross on them; or the use of coins, stamps or utensils that have a religious symbols on them, that are used in our days only as a decoration. (Shach ibid., Chochmas Odom 85,1, see also Igros Moshe Y.D. 1: 69 in regard to a medallion or an award and Igros Moshe Y.D. II, 53 regarding teaching Greek philosophy). Poskim argue that since the idols of the Babylonian month names are no longer worshiped or revered, there is no prohibition mentioning them.
Horav Shlomo Miller Shlit’a pointed out that the reason these names could be used, may be because the different idols were named after the already existing names of the months and not the opposite.
Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit’a