The first political party in the United States was known as the "Republican Party" (that's what they chose to call themselves). If you count the Federalists as a political party, it might have been the second. The name is historically correct, but, we can't even use that name anymore without creating confusion. We generally call the first "Republican Party" (the faction led by Jefferson and Madison) the Democratic Republican Party, because the modern Republican Party stole their name. The Same group has also been called the Anti Federalist Party, because they opposed the policy of the group commonly known as the Federalist Party, but this is confusing because these guys were actually federalists, and there was an earlier group which opposed ratification of the Constitution which came to be known as anti-federalists. The modern Democratic Party is actually the continuation of the original "Republican Party".
In another case, the same political faction had taken the name of Whig Party, since most Americans still fondly remembered that most American Patriots of the Revolution were "Whigs" while they were still part of the British Empire. It was like a bunch of Tories calling themselves Whigs to confuse voters.
And, of course, there's the "Patriot" movement where liberals are not welcome despite the fact that many or even most of the original American Patriots were liberals.
So, what it all really means Edit
What it means is that too many people in the US really are confused about some of the language used and understood in the rest of the World. Many of us, who aren't Republicans, understand the meaning of republican (note the lower case letter "R") in the common sense of the word, and we are republicans (again the lower case letter "R") in the common sense. But, we have to be careful using the word because Republicans (note the upper case letter "R") are constantly trying to baffle us with bullshit, or are so baffled by the bullshit they've been listening to on Fox, that they don't or can't understand common sense.