Btukfyl

I'm not an expert in politics at all, but I have been wondering about this for a while.

If Church and State are supposed to be separated, isn't it a bit of a contradiction to have "God" so strongly "embedded" in politics? I'm thinking of:

In God we Trust; official motto, printed on money.

[...] so help me God; president swearing-in.

[...] God bless America; pretty much closing every official announcement.

And so on.

How is explicitly acknowledging to believe in God, as a state, not being something that intrinsically goes against the principle of separation?

No, it isn't. This has been tested in the Federal Courts, see for example O'Hair v. Blumenthal, and Aronow v. United States. The basic reasoning is summarized in this paragraph from the Anonow case:

It is quite obvious that the national motto and the slogan on coinage and currency "In God We Trust" has nothing whatsoever to do with the establishment of religion. Its use is of a patriotic or ceremonial character and bears no true resemblance to a governmental sponsorship of a religious exercise.

People can certainly disagree based on opinion, but in context of politics and law, the mention of God isn't establishment of a state religion in the U.S.

This is a rather principialist question, and politics is a much more pragmatic field.

Philosophically, yes, a State should not acknowledge a god, there are religions without gods, agnosticism and atheism. As the State should not endorses any group, the god question should be absent from the State sphere.

Pragmatically, almost all of Americans are from a abraamic religion that share a root concept of god, they fell represented and will defend the use of this symbology, even hurting the neutrality of the State. No politician would campaign against it, as there is no practical gain and a huge practical loss for them. Unless non christians/jews/mulisms, that are prejudiced by this "state monotheism", campaign actively against it, nothing would change.

The founding fathers were largely theists and largely Christian, but from various different sects of Christianity. A substantial minority like Ben Franklin were deists.

Since they couldn't all agree which type of Christianity or even whether Christianity was preferable to a more generic deism, they put the separation of church and state provisos in to prevent one sect from taking over and banning all the others by government fiat.

There was not likely any serious intent that the idea of God would be abandoned, as evidenced by the official language you mention in your question.

Further more two of the largest groups to settle America (and have the first successful settlement) were the Puritans and the Quakers, both of which came here fleeing religious persecution. The idea of religious tolerance was probably a little more appealing in early America than most places.