You’re getting a lot of comments correctly pointing out that ARPANET was actually invented by the US in the 1970s and was the precursor to the Internet. I think it’s your question which is phrased incorrectly, and not the point you’re trying to make. Assuming this and rephrasing your question to mean the World Wide Web (not the Internet), you’re correct, that was created by Tim Berners-Lee while working at CERN in the 1990s, approximately 20 years after ARPANET. This brought along Hypertext websites, and basically was another step in the foundation of the internet as we know it today.
So rephrasing your question to “why do americans assume they invented the web (websites)?”, it’s mainly because the underlying infrastructure of the internet was originally developed by the US government, so even before websites existed, domain names were heavily American leaning, with .gov being US Government websites, and .edu being US Universities, etc. Other countries at the time had ccTLD for their country code, like .uk, .au, etc and when it came time to assign domain names, they chose to use .co.uk or .com.au for example, rather than .com.
I assume that americans rarely encounter a .com.au or other ccTLD domain names, and largely are going to .com websites. They probably assume that the .au TLD was tacked on to support Australia because they didn’t invent the internet.
the clip in all its glory: https://www.youtube.com/watch?v=FaGYXjMwS60