It's a democratic republic in the sense that we have a direct election of leaders and lawmakers. It's a bit pedantic because nowadays people use them interchangeably but we don't really have direct elections over the laws themselves (not at the federal level anyways, some states do allow for direct votes on amendments at the state and local level).

It's a spectrum, but the more weight the popular vote carries, the further towards that end it is.

People today think of the US as one "country" but when its Constitution was written it was much closer to the EU: an alliance and economic free trade and free movement area with basic shared rules.

The vote to create the United States was done by unanimity of Member States not by a popular vote.

Is the EU a "democracy"?

Athens was.

It is an eye roller for me as well.

Democracy is not something you are, it's something you do. That means it is not a black and white state, but a summation of the spectrum or beliefs, actions, and culture held by a population. When there are cultural changes that has an effect on the degree of democracy.

Here Timothy Snyder explains what the politics of inevitability is and how Walter is likely a practitioner of it and how that leads him to wrong conclusions (and there are other videos that explain Walter's American exceptionalism and why it's wrong):

https://www.youtube.com/watch?v=Eghl19elKk8

This use of republic is particular to the US. Recently, it has became particular to the Republican Party. Some of it is from the earlier usage, but I suspect it is really to contrast the Republican and Democratic parties.