They can be weird about them but they can’t deny their legitimacy. Americans do have the right to cross state borders. In other countries during Covid, the ability to cross state/regional borders was denied. It made me appreciate the decisions made when founding the country a lot more.
The problem the GP is talking about isn’t in regards to traveling and being denied. That’s not an issue in the US. The issue they would be talking about is what happens IN the state there traveling. For example a bar in Wisconsin could deny you because they don’t believe a Washington DC ID to be real.