A few disparate observations:
The original covenant relationship 'twixt God and man, with its complicated duties, loyalties, and responsibilities, was of course that between Jehovah and the Hebrew race (His “Chosen People”). Something of that spirit echoed in the Puritan theodicy, in the congregationalist organization of his church, in the covenant relationship between a parishoner and his church, and in specific creeds like justification by faith and the very concept of an Elect.
Certainly, church organizational principles differ greatly, not just between the RCC and Protestantism generally, but within different branches of the Protestant tree. Churches can be run along anything from authoritarian models to an at least partly democratic model in which the laity has a strong voice in determining policy and even in interpreting Scripture, as well as be either creedally autonomous or doctrinally subservient to a body of bishops or other high church officers. So even in the colonial era, when one’s colony might have been administered by a aristocratic client of the monarch by right of a royal charter, one might also attend a church organized along congregationalist principles, reading from the King James Version, but doctrinally responsible to no ruling body.
And to give credit where credit is due, to a large degree the political and religious freedoms in the thirteen colonies reflected and were a logical extension of the principles of religious tolerance (in particular, tolerance of Protestant Nonconformists) and of parliamentary limits to monarchical power, along with a Bill of Rights, established in England’s Glorious Revolution of 1688-90 (itself an extension of even earlier limits to monarchical absolutism established in the Magna Carta).
On the religious side, the substantial presences of Dutch Reformed in New York, Catholics in Maryland, Quakers in Pennsylvania, atheists in Rhode Island, and others (like French Huguenots) reflected both religious intolerance in the mother country and elsewhere in Europe as well as the ability for these minorities to establish themselves in the New World, often with the legitimacy conferred by a royal charter.
Politically, the Glorious Revolution provided much of the justification for the colonialists’ demands for parliamentary representation to redefine and defend our legal responsibilities and rights WRT England, and likewise influenced the post-independence demands by many of the Founding Fathers for a Bill of Rights, limiting the powers of government over the individual, to be appended to the constitution establishing a more strongly federal union of those thirteen independent states (superceding the failed initial arrangement under the Articles of Confederation).