England was at one point absolutely not secular. Secularism is tagged on after the fact and a lot of its non-secular institutions remain for ceremonial/historical reasons. Spiritual members of the house of lords (technically can be of any religion) have very very little power. If they did have any significant amount of power, I'd consider the UK a non-secular country.

POTUS can say whatever the fuck he/she wants. "God Bless America" or "Satan bless America" or "Curse this country". It is irrelevant what he/she says. If Sanders got elected, I doubt he would say "God Bless America". In court you can swear on whatever book you want, or not swear to a religious book if you don't. "In God we Trust" was added during the red scare and is very unfortunate to have it on our bills. The supreme court ruled that it was there for historical reasons, not because the government supports one belief over the other. I disagree.

Party names could be anything, doesn't mean the country is not secular.

Listen, implementation is important, Turkey technically has the freedom of press and the freedom of expression guaranteed by the Turkish constitution. The implementation of those laws however are vastly different than what you get in the west. To a point that a majority of us would argue that the freedom of expression and the freedom of press do not exist in Turkey. Now if I pointed this out, would you say "wait a minute, the NYTimes cannot print child porn! By that standard, there is no freedom of press in the US either!"? 'Cuz that's exactly what you're doing in this case.

Let me repeat: Turkey was founded with the constitution listing Islam as it's official religion. Turkey has, since its founding, had government institution that provide support for one and only one religion. It has public religious schools which teach only Islam. The government funds the celebration of Prophet Muhammed's birthday. [The government funds mass circumcision of boys.](hhttps://www.youtube.com/watch?v=SzeEQo6nH_c)