Short Answer
The question is based upon a false premise. While there are ethnic based political parties in many multi-party democracies in countries with multiple ethnicities, the mere fact that any given ethnicity tends to favor one party more than another does not make it ethnic based and the examples given in the question are flawed for making that assumption.
Long Answer
There is no multi-ethnic country with more than one political party in the world where there is no statistically significant correlation between ethnicity and party affiliation.
But the question's premise that such correlations amount to "ethic-based party politics" is not sound. And, even in countries in which some political parties have an ethnic or nationalist basis, other political parties do not.
There are certainly explicitly ethnic or nationalist based political parties. Canada has the "Parti Quebecois". The U.K. has a Scottish Nationalist Party and the Alba Party (which is also Scottish Nationalist), Plaid Cymru (a Welsh Nationalist Party), and ethnically organized political parties in Northern Ireland (divided by religious/ethnic affiliation, Democratic Unionist Party which is Protestant affiliated and associated with comparative recent (by Irish standards) English and Scottish immigration, and Sinn Féin, which is Irish Republican, ethnically indigenous Irish to times immemorial, and overwhelmingly Catholic). Spain has a Basque Nationalist party and the Catalan Nationalist Republican Left of Catalonia associated with ethnic/linguistic minorities with a strong regional identity.
Similarly, Israel, Nigeria, and Belgium, all have explicitly ethnic based political parties.
But, it isn't simple. It is also worth noting that political parties are frequently one of the key mechanism by which ethnicity and nationalist identity is formed and maintained. For example, the existence of a Finnish nation and ethnicity was strongly facilitated by Finnish Nationalist political movements and political parties. Likewise, while South Korea is currently a more or less mono-ethnic nation, the restoration of Korean sovereignty in an expansionist Japanese sphere of influence is largely a product of Korean nationalist political movements (as well as, rather oddly, the YMCA, yes, really, the Young Men Christian's Association that builds urban gyms that originally catered to young men migrating to industrial cities from rural farm communities in the U.S.A.).
But the characterization of the Tories and Labour as ethnically based doesn't stand up to close scrutiny. The ethnic and religious divisions between these parties is less clean than the ethic and religious divisions between the Democrats and Republicans in the U.S. The basis and ideology of the Tories, Labour and the Liberals, all began at a time when there were almost no non-white voters in England and historically reflect divisions among whites by social class, rather than an ethnic basis (and the U.K. is still much less ethnically diverse than the U.S. or Canada, for example). These parties aren't exactly the same in character as the parties of that early era, but they haven't experienced the wholesale realignment and trading of places that the Democrats and Republicans have in the United States and have continuity of ideology and character to an earlier date.
The rise of the ethnically based BJP party in India is a relatively recent development and the Congress party was founded as a unifying multi-ethnic party.
Likewise, there are many significant ethnic populations that do not have corresponding political parties in multi-party Democracies. Japan does not have a political party corresponding to its Korean minority. Germany does not have a political party corresponding to its Turkish minority. France does not have a political party corresponding to its Algeria or West African minorities. The U.K. does not have a political party corresponding to its South Asian minority.
The U.S. has a major political party, the Democrats, that is disproportionately favored by non-whites, but the Democratic Party is not a political party with an ethnic basis - it is a grab bag of almost all non-white ethnicities and non-Christian religions that includes liberal atheists and conservative Muslims alike (a division that arguably even explains why Cuban refugee descended populations which mostly identify as white vote Republican, and why Southern Texan Tejano Hispanics who identify as white Texans do so as well).
A multi-ethnic political party is not an ethnic based political party, and the same can be said of Labour in the U.K. and various Democratic Socialist parties in Continental Europe and Scandinavia.
The Republican Party is not yet an ethnic based political party
The U.S. Republican Party may be on its way to becoming a Christian White Nationalist Party, but this trend has been recent, gradual, and still hasn't run its course. It is not organized and defined on an ethnic basis. This is particularly true outside of the deep South where racial block voting is not pronounced.
Perhaps the GOP will have a true ethnic basis in 2040, and it is trending in that direction, but it hasn't yet crossed that line, and a lot can happen in twenty years that no one would ever predict.
While locally born white Protestant Christians in the South overwhelmingly vote for Republicans these days, this is not true of whites in other regions of the country, whites who have migrated to the South, and whites who are not Protestant or Mormon, where the white vote is very split on lines that don't cleanly correspond to ethnicity.
Conversely, Utah Mormons and white Southern Baptists in the South are two of the most loyal core supporters of the GOP but share only a smattering of religious beliefs, have little recent shared ancestry or ancestral culture (Utah Mormons are mostly ancestrally derived from white New Englanders), and speak different regional dialects of English. Utah Mormons are overwhelmingly urban. Southern Baptists in the South are disproportionately rural.
There have been a number of very prominent African-American members in the post-realignment era when the GOP was the anti-civil rights party (e.g. the recently deceased Colin Powell, U.S. Supreme Court Justice Clarence Thomas, Dale Wainwright who is a Texas Supreme Court Justice, Michael Steele who is the 64th Chairman of the Republican National Committee, J. Kenneth Blackwell who held statewide office in Ohio, Herman Cain former GOP Presidential candidate, Ben Carson who was HUD Secretary under Trump, Samuel Pierce who was a HUD Secretary, James Evans who was chair of the Utah GOP, Curtis Hill who was Indiana's Attorney General, Boyd Rutherford who is Lt. Governor of Maryland, and Alveda King who is the niece of MLK, Jr.).
Latino support for the GOP, particular among Cuban-Americans and certain South Texans was pivotal numerous recent elections. There are also notable Asian-American (including South Asian American Hindus) in the GOP, even though it is predominantly Christian.
The U.S. Republican party is less of an ethnic party than, for example, the Christian Democratic Union in Germany.
