Washington was apparently emphasizing what he saw as the commonalities among the people and in their religious beliefs.
But at the time, those differences were considerable, and led even to different Protestant groups persecuting each other. When I have more time, I can post more about that, but suffice to say that there were genuine and deep problems and differences among Puritans, Quakers, Baptists, Lutherans, and Anglicans, etc., and there were many Catholics and some Jews here as well.
And even where there was some acceptance to allow those of different sects to have their own churches and not try to interfere with them, that doesn’t mean that Lutherans and Baptists would have accepted an Anglican state religion established by Congress, or Anglicans and Puritans would have accepted a Baptist state religion established by Congress, etc. Certainly in establishing a Union, that was the danger, that one sect would grow the most powerful and establish itself as the national religion, if it were allowed to, and that would have been a public concern of the time.
Yes. For that reason, Congress was prohibited from promoting a particular sect, but States were not prohibited.
The states had certain things they were not permitted to do. Those were articulated in Article One, Section Ten. Establishing a State religion was not prohibited.