In spanish we have the word "indio" which means indian exactly like in english (both for the subcontinent and the americas) and "indiano" which can refer to anything (people, art or whatever) coming from the iberoamerican colonies (that is, also from colonial times. It's not used generally for post-independency period) Since the word "indio" causes confussion, at least in Spain often people uses the term "hindú" for people from the subcontinent, so a mislead leading to another misleading term...
In German, we still use the word Indianer today to describe native Americans.
But the term has no negative connotation, unlike the use English term Indian for native Americans apparently has.
Mandarin uses a similar division, with "Yin du" to denote India and its people and "Yin di an," a loanword from English, for the indigenous people of the Americas. Indian people and the Hindu religion are distinguished by the character 人 (ren) for person or 教 (jiao) for teaching or religion tacked on to the end of 印度 (yin du).
Anyway, I tend to think Native American is just as geographically confusing as American Indian, given the tendency for many people around the world to associate "American" with the United States specifically rather than the Americas in general. People tend to associate Native American with the indigenous people of the US and overlook the indigenous peoples of Canada and Latin America, who stem from the same founding population and in some cases, like the Mohawk, Tlingit, and Tohono Oodham, continue to transcend international borders to this day.
"American Indian" also has this problem, but people often keep the Indian part and tack on other countries or regions - Caribbean Indian, Latin American Indian (used in the US Census), or Canadian Indian.
I've heard that "First Nations" is becoming more common in the Pacific Northwestern US, but I don't have any firsthand experience. When I was in Ecuador, "indigena" seemed to be supplanting "indio" as the most acceptible term in mainstream circles there, but I can't attest to other Spanish-speaking countries.
In Europe, there are 234 extant languages; if we were to extant the same lax criteria of 'mutual intelligibility' which we use to define Amerindian languages (such as Quechua, Mayan and Aymara) from their respective 'dialects', then this would be whittled down to only 16 extant European languages- 9, excluding those 7 extant languages of which aren't actually European in origin. The only extant European languages by this token would Slavic, Romance, Germanic, Celtic, Greek, Baltic, Albanian, Armenian and Basque. Would this seem fair to you?
Am I missing something? Why wouldn't Finnic, Sami, Mari, Mordvin, Permic, and Samoyed be considered native to Europe? Or, if we're counting Armenia as European, the various Caucasian languages like Georgian, Abkhaz, and Chechen? Or Ossetian, the only surviving descendant of the Scythian languages that dominated southeastern Russia before Slavic? I'm curious as to what your other seven non-native languages would be, as my count of Hungarian, Maltese, Romani, Kalmyk, and at least three different Turkic branches in addition to the others mentioned numbers more than that.