Country in the world or county in the country?
Raising Arizona is one of my favorite films.
"Estadounidense" is the demonym--so what you would call something from the US (the English equivalent would be American, possibly Yankee [although that has its own Spanish word, "yanqui"]). Other demonyms would be salvadoreño for Salvadorean, mexicano for Mexican, venezolano for Venezuelan, etc.
So, to answer your question: yes, the words are related; someone from los Estados Unidos (EEUU) would be estadounidense.
Edit to clarify:
Strictly speaking, the word "demonym" refers to people, but in the case of "estadounidense" it can refer to things and people. From English Wikipedia:
"Often, demonyms are the same as the adjectival form of the place, e.g. Egyptian, Japanese, or Greek. However, they are not necessarily the same, as exemplified by Spanish instead of Spaniard or British instead of Briton."
My goodness, I had forgotten all about Bug!
My Sonic CD disc was different: it was printed with Sonic rolled up in a ball so he spun around in your CD drive. I scratched that CD up bad when I closed the disc tray and it wasn't set in it correctly...