From the Wikipedia page:
Secular states do not have a state religion (e.g. an established religion) or an equivalent, although the absence of an established state religion does not necessarily mean that a state is fully secular in all respects. For example, many secular states have religious references in their national anthems and flags.
The US have many more religious references than just the dollar bill. The president has to swear on the bible. Court witnesses have to swear before God.
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