In the Bavli to that Mishna (Gittin 80a), 'Ula explains that the reason there was an enactment made to write the date according to the local government was "משום שלום מלכות" "to maintain peace with the government". Rashi explains that the governments would see we use their dating system and assume that we value their leadership. Accordingly, Rambam rules (Hilchot Geirushin 1:27 (English)):
וכן תיקנו שיהיו מונין בגיטין למלכות אותו הזמן, משום שלום מלכות. כתב למלכות שאינה מלכות אותה המדינה, או לבניין הבית, או לחורבן הבית--אם דרך אנשי אותו מקום למנות בו, הרי זה כשר; ואם אין דרכן למנות בו, הרי זה פסול. וכבר נהגו כל ישראל למנות בגיטין, או ליצירה או למלכות אלכסנדרוס מקרון שהוא מניין שטרות. ואם כתב לשם מלכות אותו זמן במדינה שיש בה רשות אותה מלכות, הרי זה כשר.
Similarly, [our Sages] ordained that the year of the ruling kingdom of that time should be mentioned in a get to gain the favor of the ruling authorities.
[The following rules apply if] a person writes a get and dated it according to the years of a kingdom other [than that of his locale] or according to the years beginning from the Temple's construction or destruction. If it is customary for people in that locale to date [their documents] in this manner, it is acceptable. If this is not the local custom, it is unacceptable.
It has already become the universal Jewish custom to date gittin from the time of creation, or from the crowning of Alexander the Great, which is [the accepted means of dating] for legal documents. If one dates [a get] according to the years of a contemporary kingdom, it is acceptable only in the country over which that kingdom rules.
It seems that now that governments don't care how we date the documents, it is acceptable to use a local customary system.
