Church and State are legally separated in Mexico. When that happened during the Reforma period after the revolution, it resulted in repression of the Church and the resulting Cristero war. In the end, the government seized all Church property. Which means that in a land where Church and State are legally separated, the government owns and maintains all Church properties! And Christmas shows up all over the place, from official squares to nativity scenes to toll booths (festooned with garland and “Feliz Navidad” written on the windows).
Feliz Navidad remains a common greeting, although a few “feliz fiestas” or “happy holidays” have crept into Méxican culture. Since “feliz fiestas” sounds so weird (are there any unhappy fiestas?) it may not catch on.
Mexico has a tortured history of religious involvement in political affairs. For a long time, all citizens had to be Catholic. The call for independence came under a banner of the Virgin of Guadalupe. Still, México seems to have come to terms with a way to be officially secular without renouncing the essence of religious belief, which is after all communal and “other” facing.
¡Feliz Navidad a todos!