Same reason the Democrats call themselves Democrats. These names stem from "Republic" and "Democracy, two things most Americans have positive feelings toward. Each party wants to be seen as very patriotic, and using names like those helps their cause.
Many Democrats on the west coast of the United States of America do see the anti-slavery politician Tomas Jefferson who lived long ago as their hero. His party was called the Democratic-Republican Party. They say their party has the same values, and that they shortened the name to Democrat when the Democratic-Republican Party began to work with other Democrats. They often meet with and debate people who call themselves the Republican Party who also are a group of people who say they have Jefferson's values and are like the Democratic-Republican Party.
They call themselves the Diné. :)
They call themselves Rastas.
I don't know what they called themselves, but they we originally referred to as "Anti-Federalists". Later as "Republicans". After that a variety of Republican-Democrats, Jefferson-Democrats, Jeffersonians, etc. Finally, Democrats - a named that has remained.
they call themselves tamalians
They called themselves catrachos and argentinian cal themselves argentinians
They call themselves Evangelicals
They call themselves Muskogee, or Muscogulge
No. They call themselves Muslim, but not Arabs.
A Republican.
Germans call themselves "Deutsche" in their native language.
In the United States, the Republican Party has a larger percentage of people who identify themselves as members compared to the Democratic Party.