How can we and why do we?
The Ramban דרשה לר"ה explains why Klal Yisroel adopted Babylonian names for the months.*
תמוז is a name of an עבודה זרה
יחזקאל פ"ח הנשים ישבות מבכות את התמוז
Many attempted to answer how we may and why we did name the month after an Avoda Zara.
Harav Hagaon R. Chaim Kanievsky Shlita.
said, being it was the month they placed a צלם in the היכל
I wasn't זוכה to understand any of the answers (includings R. Chaim's).
*Originally the months had no names
but numbers 1st 2nd etc. except for
אביב -זיו - בול -ירח האתנים