Switzerland is a republic too. Most democracies are either republics or constitutional monarchies.

There is no contradiction between being a republic and a democracy unless one unreasonably restricts the definition of democracy to direct democracy. Reasonable definitions of democracy include representative democracy, which is practiced in many republics.

Democracy and Republic aren't mutually exclusive. I'd call rule of law with a democratic constitution (written or unwritten), political representation (direct or indirect), respect of human rights and civil rights to be the elements of modern liberal democracy.